Here_are_the.pdf

Here are the 14 MCQs with the answers placed at the end: 1. If 𝐴 is a square matrix, which of the following is true?  (A) 𝐴(adj𝐴) =∣ 𝐴 ∣ 𝐼𝑛 = (adj𝐴)𝐴  (B) 𝐴(adj𝐴) = 𝐼𝑛  (C) 𝐴(adj𝐴) = (adj𝐴)𝐴  (D) 𝐴 =∣ 𝐴 ∣ 𝐼𝑛 2. What is the value of ∣ adj𝐴 ∣ for a square matrix 𝐴?  (A) ∣ 𝐴...

Here are the 14 MCQs with the answers placed at the end: 1. If 𝐴 is a square matrix, which of the following is true?  (A) 𝐴(adj𝐴) =∣ 𝐴 ∣ 𝐼𝑛 = (adj𝐴)𝐴  (B) 𝐴(adj𝐴) = 𝐼𝑛  (C) 𝐴(adj𝐴) = (adj𝐴)𝐴  (D) 𝐴 =∣ 𝐴 ∣ 𝐼𝑛 2. What is the value of ∣ adj𝐴 ∣ for a square matrix 𝐴?  (A) ∣ 𝐴 ∣𝑛  (B) ∣ 𝐴 ∣𝑛−1  (C) ∣ 𝐴 ∣2  (D) ∣ 𝐴 ∣−1 3. For a square matrix 𝐴, what is the formula for adj(adj𝐴)?  (A) 𝐴2  (B) (adj𝐴)2  (C) ∣ 𝐴 ∣𝑛−2 𝐴  (D) ∣ 𝐴 ∣ 𝐴 4. What is the formula for ∣ adj(adj𝐴) ∣?  (A) ∣ 𝐴 ∣𝑛−1  2  (B) ∣ 𝐴 ∣(𝑛−1)  (C) ∣ 𝐴 ∣2  (D) ∣ 𝐴 ∣𝑛−2 5. Which of the following is true for adjugates of transposed matrices?  (A) adj(𝐴𝑇 ) = 𝐴𝑇  (B) adj(𝐴𝑇 ) = adj𝐴  (C) adj(𝐴𝑇 ) = 𝐴−1  (D) adj(𝐴𝑇 ) = (adj𝐴)𝑇 6. For two square matrices 𝐴 and 𝐵, which of the following is true?  (A) adj(𝐴𝐵) = adj𝐴 ⋅ adj𝐵  (B) adj(𝐴𝐵) = (adj𝐴)(adj𝐵)  (C) adj(𝐴𝐵) = (adj𝐵)(adj𝐴)  (D) adj(𝐴𝐵) = (adj𝐴)𝑇 (adj𝐵) 7. For any natural number 𝑚, what is the formula for adj(𝐴𝑚 )?  (A) 𝐴𝑚  (B) (adj𝐴)𝑚  (C) 𝐴𝑚−1  (D) (adj𝐴)−𝑚 8. If 𝑘 ∈ ℝ and 𝐴 is a square matrix, what is adj(𝑘𝐴)?  (A) 𝑘 𝑛 ⋅ adj𝐴  (B) 𝑘 𝑛−1 ⋅ adj𝐴  (C) 𝑘 ⋅ 𝐴  (D) 𝑘 𝑛+1 ⋅ adj𝐴 9. What is the adjugate of the identity matrix 𝐼𝑛 ?  (A) 𝐼𝑛  (B) 𝐴  (C) 0  (D) 𝑛 ⋅ 𝐼𝑛 10. What is adj(0)?  (A) 0  (B) 𝐴  (C) 𝐼𝑛  (D) 𝑛 11. If 𝐴 is a symmetric matrix, which of the following is true about adj𝐴?  (A) adj𝐴 is skew-symmetric  (B) adj𝐴 is diagonal  (C) adj𝐴 is also symmetric  (D) adj𝐴 is triangular 12. If 𝐴 is a diagonal matrix, which of the following is true about adj𝐴?  (A) adj𝐴 is zero  (B) adj𝐴 is also diagonal  (C) adj𝐴 is symmetric  (D) adj𝐴 is triangular 13. If 𝐴 is a triangular matrix, what is true about adj𝐴?  (A) adj𝐴 is symmetric  (B) adj𝐴 is also triangular  (C) adj𝐴 is diagonal  (D) adj𝐴 = 𝐴 14. Which of the following is true if 𝐴 is a singular matrix?  (A) ∣ adj𝐴 ∣=∣ 𝐴 ∣  (B) ∣ adj𝐴 ∣= 1  (C) ∣ adj𝐴 ∣= 0  (D) ∣ adj𝐴 ∣=∣ 𝐴 ∣−1 Answers: 1. (A) 2. (B) 3. (C) 4. (B) 5. (D)  6. (C) 7. (B) 8. (B) 9. (A) 10. (A) 11. (C) 12. (B) 13. (B) 14. (C) Here are 7 MCQs, one for each formula related to invertible matrices: 1. If 𝐴 is an invertible matrix, what is (𝐴−1 )−1 ?  (A) 𝐴−2  (B) 𝐴−1  (C) 𝐴2  (D) 𝐴 2. Which of the following is true about the inverse of the transpose of a matrix?  (A) (𝐴𝑇 )−1 = 𝐴𝑇  (B) (𝐴𝑇 )−1 = (𝐴−1 )𝑇  (C) (𝐴𝑇 )−1 = (𝐴−1 )−𝑇  (D) (𝐴𝑇 )−1 = 𝐴 3. For two invertible matrices 𝐴 and 𝐵, which of the following is true for their product?  (A) (𝐴𝐵)−1 = 𝐴−1 𝐵−1  (B) (𝐴𝐵)−1 = (𝐴𝐵)  (C) (𝐴𝐵)−1 = 𝐵 −1 𝐴−1  (D) (𝐴𝐵)−1 = 𝐵𝐴 4. Which of the following is correct for the inverse of a matrix power 𝐴𝑘 , where 𝑘 ∈ ℕ?  (A) (𝐴𝑘 )−1 = (𝐴−1 )𝑘  (B) (𝐴𝑘 )−1 = (𝐴)𝑘+1  (C) (𝐴𝑘 )−1 = 𝐴−𝑘+1  (D) (𝐴𝑘 )−1 = 𝐴−𝑘 5. If 𝐴 is an invertible matrix, what is adj(𝐴−1 )?  (A) adj(𝐴)  (B) (adj𝐴)−1  (C) (adj𝐴)𝑇  (D) adj(𝐴𝑇 ) 6. For any invertible matrix 𝐴, which of the following is true about its determinant?  (A) ∣ 𝐴−1 ∣= 1/∣ 𝐴 ∣=∣ 𝐴 ∣−1  (B) ∣ 𝐴−1 ∣=∣ 𝐴 ∣  (C) ∣ 𝐴−1 ∣= 0   (D) ∣ 𝐴−1 ∣= −∣ 𝐴 ∣ 7. If 𝐴 is a diagonal matrix 𝐴 = diag(𝑎1 , 𝑎2 , … , 𝑎𝑛 ), what is 𝐴−1 ?  (A) 𝐴−1 = diag(𝑎1−1 , 𝑎2−1 , … , 𝑎𝑛−1 )  (B) 𝐴−1 = diag(𝑎1 , 𝑎2 , … , 𝑎𝑛 )  (C) 𝐴−1 = diag(𝑎12 , 𝑎22 , … , 𝑎𝑛2 )  (D) 𝐴−1 = 𝐼𝑛 Answers: 1. (D) 2. (B) 3. (C) 4. (A) 5. (B) 6. (A) 7. (A) Here are 5 MCQs based on the properties of matrix multiplication: 1. Which of the following is generally true about matrix multiplication?  (A) 𝐴𝐵 = 𝐵𝐴  (B) 𝐴𝐵 ≠ 𝐵𝐴  (C) 𝐴 + 𝐵 = 𝐵 + 𝐴   (D) 𝐴𝐵 = 𝐴 2. Which of the following matrix types allows for commutative multiplication, i.e., 𝐴𝐵 = 𝐵𝐴?  (A) Any random matrices  (B) Identity or diagonal matrices  (C) Symmetric matrices  (D) Lower triangular matrices 3. Which property of matrix multiplication states that 𝐴(𝐵 + 𝐶 ) = 𝐴𝐵 + 𝐴𝐶?  (A) Commutativity  (B) Distributivity  (C) Associativity  (D) Symmetry 4. The associativity property of matrix multiplication implies:  (A) (𝐴𝐵)𝐶 = 𝐴(𝐵𝐶 )  (B) 𝐴𝐵 = 𝐵𝐴  (C) (𝐴 + 𝐵)𝐶 = 𝐴(𝐶 + 𝐵)  (D) 𝐴(𝐵 + 𝐶 ) = 𝐴𝐵 + 𝐴𝐶 5. Which of the following expressions demonstrates the distributive property of matrix multiplication?   (A) (𝐴𝐵)𝐶 = 𝐴(𝐵𝐶 )  (B) 𝐴(𝐵 + 𝐶 ) = 𝐴𝐵 + 𝐴𝐶  (C) 𝐴𝐵 = 𝐵𝐴  (D) (𝐴 + 𝐵)(𝐶 + 𝐷) = 𝐴𝐶 + 𝐵𝐷 Answers: 1. (B) 2. (B) 3. (B) 4. (A) 5. (B) Here are the 5 MCQs with the correct options distributed randomly: 1. Which of the following is generally true about matrix multiplication?  (A) 𝐴𝐵 = 𝐵𝐴  (B) 𝐴𝐵 ≠ 𝐵𝐴  (C) 𝐴 + 𝐵 = 𝐵 + 𝐴  (D) 𝐴𝐵 = 𝐴 2. Which of the following matrix types allows for commutative multiplication, i.e., 𝐴𝐵 = 𝐵𝐴?  (A) Any random matrices  (B) Identity or diagonal matrices   (C) Symmetric matrices  (D) Lower triangular matrices 3. Which property of matrix multiplication states that 𝐴(𝐵 + 𝐶 ) = 𝐴𝐵 + 𝐴𝐶?  (A) Commutativity  (B) Distributivity  (C) Associativity  (D) Symmetry 4. The associativity property of matrix multiplication implies:  (A) (𝐴𝐵)𝐶 = 𝐴(𝐵𝐶 )  (B) 𝐴𝐵 = 𝐵𝐴  (C) (𝐴 + 𝐵)𝐶 = 𝐴(𝐶 + 𝐵)  (D) 𝐴(𝐵 + 𝐶 ) = 𝐴𝐵 + 𝐴𝐶 5. Which of the following expressions demonstrates the distributive property of matrix multiplication?  (A) (𝐴𝐵)𝐶 = 𝐴(𝐵𝐶 )  (B) 𝐴(𝐵 + 𝐶 ) = 𝐴𝐵 + 𝐴𝐶  (C) 𝐴𝐵 = 𝐵𝐴  (D) (𝐴 + 𝐵)(𝐶 + 𝐷) = 𝐴𝐶 + 𝐵𝐷 Answers:  1. (B) 2. (C) 3. (B) 4. (A) 5. (B) Here are 5 MCQs based on the properties of the transpose of a matrix: 1. What is the result of taking the transpose of the transpose of a matrix 𝐴?  (A) 𝐴2  (B) 𝐴  (C) 𝐴′  (D) 2𝐴′ 2. If 𝑘 is a scalar and 𝐴 is a matrix, what is the expression for the transpose of 𝑘𝐴?  (A) 𝑘𝐴  (B) 𝐴′  (C) 𝑘𝐴′  (D) 𝐴 3. Which property of the transpose states that the transpose of the sum of two matrices equals the sum of their transposes?   (A) (𝐴 + 𝐵)′ = 𝐴′ + 𝐵′  (B) (𝐴 + 𝐵)′ = 𝐴′ − 𝐵′  (C) (𝐴 + 𝐵)′ = 𝐴𝐵′  (D) (𝐴 + 𝐵)′ = (𝐴′ + 𝐵) 4. What is the result of the transpose of the product of two matrices 𝐴 and 𝐵?  (A) 𝐴𝐵′  (B) (𝐴𝐵)′  (C) 𝐵′ 𝐴′  (D) (𝐴′ 𝐵′ ) 5. For three matrices 𝐴, 𝐵, and 𝐶, what is the expression for the transpose of the product 𝐴𝐵𝐶?  (A) 𝐶 ′ 𝐵′ 𝐴′  (B) 𝐴𝐵𝐶 ′  (C) (𝐴𝐵𝐶 )′  (D) 𝐴′ 𝐶 ′ 𝐵′ Answers: 1. (B) 2. (C) 3. (A) 4. (C) 5. (A) Here are 5 MCQs based on the definition and properties of the trace of a matrix: 1. What is the definition of the trace of a matrix 𝐴?  (A) The sum of all elements in 𝐴  (B) The product of the diagonal elements of 𝐴  (C) The sum of the diagonal elements of 𝐴  (D) The average of the diagonal elements of 𝐴 2. If 𝑝 is a scalar and 𝐴 is a square matrix, what is the expression for the trace of 𝑝𝐴?  (A) Tr(𝑝𝐴) = 𝑝Tr(𝐴)  (B) Tr(𝑝𝐴) = 𝑝 + Tr(𝐴)  (C) Tr(𝑝𝐴) = 𝑝 ⋅ 𝐴  (D) Tr(𝑝𝐴) = Tr(𝐴) 3. If 𝐴 and 𝐵 are square matrices of the same size, which property of the trace can be expressed as Tr(𝑝𝐴 + 𝑞𝐵) = 𝑝Tr(𝐴) + 𝑞Tr(𝐵)?  (A) Cyclic property  (B) Linearity  (C) Scalar multiplication  (D) Non-linearity 4. What does the cyclic property of the trace state about two square matrices 𝐴 and 𝐵?  (A) Tr(𝐴𝐵) = Tr(𝐵𝐴)  (B) Tr(𝐴𝐵) = Tr(𝐴) ⋅ Tr(𝐵)  (C) Tr(𝐴𝐵) + Tr(𝐵𝐴) = 0  (D) Tr(𝐴 + 𝐵) = Tr(𝐴) + Tr(𝐵) 5. Which of the following statements about the trace of a matrix is true?  (A) The trace is defined only for rectangular matrices.  (B) The trace of a matrix is always zero.  (C) The trace of a matrix depends on its eigenvalues.  (D) The trace is invariant under cyclic permutations of matrix products. Answers: 1. (C) 2. (A) 3. (B) 4. (A) 5. (D) Here’s a similar MCQ based on the note about the trace of the matrix 𝐴𝐴𝑇 or 𝐴𝑇 𝐴: 1. What does the trace of the matrix product 𝐴𝐴𝑇 or 𝐴𝑇 𝐴 represent?   (A) The sum of the diagonal elements of 𝐴  (B) The product of all elements of 𝐴  (C) The sum of the squares of all the elements of 𝐴  (D) The determinant of the matrix 𝐴 Answer: 1. (C) Here are 5 MCQs based on the definitions and properties of symmetric and skew-symmetric matrices: 1. What is the condition for a square matrix 𝐴 to be classified as symmetric?  (A) 𝐴′ = −𝐴  (B) 𝐴 = 0  (C) 𝐴′ = 𝐴  (D) 𝐴′ = 2𝐴 2. Which of the following is true about the diagonal elements of a skew-symmetric matrix?  (A) All diagonal elements are equal.  (B) All diagonal elements are zero.  (C) All diagonal elements are positive.  (D) All diagonal elements are negative. 3. If 𝐴 is a square matrix, what can be said about the sum 𝐴 + 𝐴′ ?  (A) It is always symmetric.  (B) It is always skew-symmetric.  (C) It is always a zero matrix.  (D) It is always singular. 4. Which statement is true regarding any square matrix 𝐴 in relation to symmetric and skew-symmetric matrices?  (A) 𝐴 cannot be expressed as a sum of symmetric and skew-symmetric matrices.  (B) Any square matrix can be written as the sum of a symmetric matrix and a skew-symmetric matrix.  (C) The sum of symmetric and skew-symmetric matrices results in a skew-symmetric matrix.  (D) The transpose of a symmetric matrix is always skew-symmetric. 5. For a symmetric matrix 𝐴, which of the following holds true?  (A) 𝐴′ = −𝐴  (B) 𝐴′ is not equal to 𝐴  (C) 𝐴 = 𝐴′  (D) All off-diagonal elements are equal. Answers:  1. (C) 2. (B) 3. (A) 4. (B) 5. (C) Here are 6 MCQs based on the definitions of special types of matrices: 1. What condition must a square matrix 𝐴 satisfy to be classified as a skew-symmetric matrix?  (A) 𝐴 = 𝐴′  (B) 𝐴′ = −𝐴  (C) 𝐴2 = 𝐴  (D) 𝐴2 = 𝐼 2. Which of the following statements is true for an orthogonal matrix 𝐴?  (A) 𝐴′ 𝐴 = 𝐼  (B) 𝐴2 = 𝐴  (C) 𝐴′ = −𝐴  (D) 𝐴2 = 0 3. A matrix 𝐴 is said to be idempotent if:  (A) 𝐴′ = −𝐴  (B) 𝐴2 = 𝐼   (C) 𝐴2 = 𝐴  (D) 𝐴2 = 0 4. For a square matrix 𝐴 to be classified as involutory, it must satisfy the condition:  (A) 𝐴2 = 𝐴  (B) 𝐴 = 0  (C) 𝐴2 = 𝐼  (D) 𝐴′ = 𝐴 5. A periodic matrix 𝐴 with period 𝑘 satisfies which of the following conditions?  (A) 𝐴𝑘 = 0  (B) 𝐴𝑘+1 = 𝐴  (C) 𝐴′ = 𝐴  (D) 𝐴2 = 𝐴 6. A square matrix 𝐴 is called nilpotent if:  (A) There exists a positive integer 𝑘 such that 𝐴𝑘 = 0  (B) 𝐴′ = 𝐴  (C) 𝐴2 = 𝐴  (D) 𝐴 = 𝐼 Answers: 1. (B)  2. (A) 3. (C) 4. (C) 5. (B) 6. (A) Here are 3 MCQs based on the notes on the determinant, minor, and cofactor of a matrix: 1. What is the determinant of a matrix?  (A) A vector  (B) A polynomial  (C) A number  (D) A function 2. How is the minor of an element [𝑎𝑖𝑗 ] defined in a matrix 𝐴?  (A) It is obtained by removing the 𝑖-th row and 𝑗-th column of the matrix 𝐴.  (B) It is the product of the diagonal elements of 𝐴.  (C) It is the sum of all elements of 𝐴.  (D) It is the same as the determinant of 𝐴. 3. What is the formula to calculate the cofactor 𝐶𝑖𝑗 of an element [𝑎𝑖𝑗 ] in a matrix?  (A) 𝐶𝑖𝑗 = 𝑀𝑖𝑗   (B) 𝐶𝑖𝑗 = (−1)𝑖+𝑗 ⋅ 𝑀𝑖𝑗  (C) 𝐶𝑖𝑗 =∣ 𝐴 ∣  (D) 𝐶𝑖𝑗 = 𝑀𝑖𝑗 2 Answers: 1. (C) 2. (A) 3. (B) Here are 7 MCQs based on the information about singular and non-singular matrices: 1. What is a singular matrix?  (A) A square matrix whose determinant is zero.  (B) A matrix with all elements equal to one.  (C) A matrix that is not invertible.  (D) A matrix with only non-negative elements. 2. A non-singular matrix is defined as:  (A) A matrix that has at least one zero element.  (B) A square matrix whose determinant is non-zero.  (C) A matrix with all elements equal to zero.  (D) A matrix with identical rows. 3. Which of the following statements is true regarding the determinant of a matrix?   (A) The determinant can be computed for non-square matrices.  (B) If any row or column of a square matrix is zero, its determinant is zero.  (C) The determinant is always a positive number.  (D) The determinant of a matrix is equal to the sum of its elements. 4. If two rows of a square matrix are equal, what can be said about its determinant?  (A) It will be equal to 1.  (B) It will be the same as the product of the diagonal elements.  (C) Its determinant will be zero.  (D) Its determinant will be non-zero. 5. What happens to the value of the determinant when two rows of a matrix are interchanged?  (A) The value of the determinant becomes zero.  (B) The value of the determinant remains the same.  (C) The value of the determinant changes by a minus sign.  (D) The determinant becomes a positive number. 6. For a diagonal matrix, how is the determinant calculated?   (A) It is the sum of the diagonal elements.  (B) It is the product of the diagonal elements.  (C) It is the average of the diagonal elements.  (D) It is the maximum diagonal element. 7. If a square matrix has a determinant of 5, what can be said about the matrix?  (A) It is a singular matrix.  (B) It is a non-singular matrix.  (C) It must be a diagonal matrix.  (D) It must have at least one zero row. Answers: 1. (A) 2. (B) 3. (B) 4. (C) 5. (C) 6. (B) 7. (B) Here are 7 MCQs based on the properties of determinants: 1. When multiplying a scalar with a determinant, how is the operation performed?   (A) Multiply the scalar with any one row or column of the matrix.  (B) Multiply the scalar with the entire matrix.  (C) Add the scalar to the diagonal elements of the matrix.  (D) Multiply the scalar with the determinant directly. 2. What is the determinant of the product of two matrices 𝐴 and 𝐵?  (A) ∣ 𝐴𝐵 ∣=∣ 𝐴 ∣ +∣ 𝐵 ∣  (B) ∣ 𝐴𝐵 ∣=∣ 𝐴 ∣×∣ 𝐵 ∣  (C) ∣ 𝐴𝐵 ∣=∣ 𝐴 ∣ −∣ 𝐵 ∣  (D) ∣ 𝐴𝐵 ∣=∣ 𝐴 ∣/∣ 𝐵 ∣ 3. How is the determinant of a matrix 𝐴 raised to a power 𝑛 calculated?  (A) ∣ 𝐴𝑛 ∣=∣ 𝐴 ∣𝑛  (B) ∣ 𝐴𝑛 ∣=∣ 𝐴 ∣𝑛−1  (C) ∣ 𝐴𝑛 ∣= 𝑛 ∣ 𝐴 ∣  (D) ∣ 𝐴𝑛 ∣=∣ 𝐴 ∣ 4. Which of the following is true about the determinant of the transpose of a matrix 𝐴?  (A) ∣ 𝐴′ ∣= −∣ 𝐴 ∣  (B) ∣ 𝐴′ ∣=∣ 𝐴 ∣  (C) ∣ 𝐴′ ∣=∣ 𝐴 ∣2   (D) ∣ 𝐴′ ∣= 0 5. What is the formula for the determinant of a scalar multiple 𝑘𝐴 of a matrix 𝐴 of order 𝑛?  (A) ∣ 𝑘𝐴 ∣= 𝑘 ×∣ 𝐴 ∣  (B) ∣ 𝑘𝐴 ∣= 𝑘 ∣ 𝐴 ∣𝑛  (C) ∣ 𝑘𝐴 ∣= 𝑘 𝑛 ×∣ 𝐴 ∣  (D) ∣ 𝑘𝐴 ∣= 𝑘 2 ×∣ 𝐴 ∣ 6. What is the determinant of a skew-symmetric matrix of odd order?  (A) Equal to the sum of diagonal elements.  (B) A non-zero constant.  (C) Equal to the product of all the matrix elements.  (D) Zero. 7. What is the determinant of the identity matrix 𝐼 of any order?  (A) 1  (B) 0  (C) It depends on the elements of the matrix.  (D) It is equal to the sum of diagonal elements. Answers: 1. (A)  2. (B) 3. (A) 4. (B) 5. (C) 6. (D) 7. (A) Here are 4 MCQs based on the provided information about the product of two matrices resulting in a zero matrix: 1. If the product of two matrices 𝐴 and 𝐵 is the zero matrix 𝐴𝐵 = 𝑂, which of the following is always true?  (A) 𝐴 must be a null matrix.  (B) 𝐵 must be a null matrix.  (C) One or both matrices are singular.  (D) Both 𝐴 and 𝐵 must be identity matrices. 2. What is a possible scenario if the product of two matrices 𝐴 and 𝐵 equals the zero matrix 𝐴𝐵 = 𝑂?  (A) 𝐴 is a zero matrix.  (B) 𝐴 is an identity matrix.  (C) 𝐴 is a diagonal matrix.  (D) 𝐴 is a non-singular matrix. 3. In the case where 𝐴𝐵 = 𝑂, but neither 𝐴 nor 𝐵 is a zero matrix, what can be concluded about 𝐴 and 𝐵?   (A) Both matrices are invertible.  (B) One of the matrices is an identity matrix.  (C) Both matrices must be diagonal matrices.  (D) Both matrices are singular. 4. If 𝐴 and 𝐵 are non-null matrices but their product 𝐴𝐵 = 𝑂, what must be true about their determinants?  (A) ∣ 𝐴 ∣= 1 and ∣ 𝐵 ∣= 1  (B) ∣ 𝐴 ∣= 0 and ∣ 𝐵 ∣= 0  (C) ∣ 𝐴 ∣= −∣ 𝐵 ∣  (D) ∣ 𝐴 ∣≠ 0 Answers: 1. (C) 2. (A) 3. (D) 4. (B) Here are 2 MCQs based on the information about the adjoint matrix and the result related to determinants: 1. What is the adjoint matrix Adj(𝐴) of a square matrix 𝐴?  (A) The transpose of the cofactor matrix of 𝐴.  (B) The inverse of the matrix 𝐴.  (C) The cofactor matrix of 𝐴.  (D) The product of matrix 𝐴 and its determinant. 2. What happens if the elements of a matrix are combined incorrectly with their cofactors?  (A) The result is the inverse of the matrix.  (B) The result will always be zero.  (C) The result will be the identity matrix.  (D) The result is the adjoint of the matrix. Answers: 1. (A) 2. (B) Here is 1 MCQ based on the shortcut for finding the adjoint of a 2x2 matrix: 𝑎 𝑏 1. Given the matrix 𝐴 = [ ], which of the following is 𝑐 𝑑 the adjoint adj(𝐴)? 𝑑 𝑏  (A) [ ] 𝑐 𝑎 𝑑 −𝑏  (B) [ ] −𝑐 𝑎 𝑎 −𝑏  (C) [ ] −𝑐 𝑑 𝑎 𝑏  (D) [ ] −𝑑 −𝑐 Answer:  1. (B) Here are 3 MCQs based on the information about the inverse of a square matrix: 1. For an 𝑛 × 𝑛 square matrix 𝐴, what condition must hold for 𝐵 to be called the inverse of 𝐴?  (A) 𝐴𝐵 = 𝑂𝑛  (B) 𝐴𝐵 = 𝐵𝐴 = 𝐼𝑛  (C) 𝐴𝐵 = 𝐼𝑛 but 𝐵𝐴 ≠ 𝐼𝑛  (D) 𝐴2 = 𝐵 2. The inverse of a square matrix 𝐴 exists if and only if:  (A) The determinant of 𝐴 is non-zero.  (B) The adjoint of 𝐴 is equal to 𝐴.  (C) All elements of 𝐴 are positive.  (D) 𝐴 is a diagonal matrix. 3. The formula for the inverse of a matrix 𝐴, where ∣ 𝐴 ∣≠ 0, is given by:  (A) 𝐴−1 =∣ 𝐴 ∣⋅ adj(𝐴)  (B) 𝐴−1 = adj(𝐴) 1  (C) 𝐴−1 = ⋅ adj(𝐴) ∣𝐴∣  (D) 𝐴−1 = 𝐴′ Answers: 1. (B) 2. (A) 3. (C) Here is 1 MCQ based on the shortcut for finding the inverse of a 2x2 matrix: 𝑎 𝑏 1. Given the matrix 𝐴 = [ ], which of the following 𝑐 𝑑 represents the inverse 𝐴−1 ? 1 𝑑 −𝑏  (A) ⋅[ ] 𝑎𝑑+𝑏𝑐 −𝑐 𝑎 1 𝑎 −𝑏  (B) ⋅[ ] 𝑎𝑏−𝑐𝑑 −𝑐 𝑑 1 𝑑 −𝑏  (C) ⋅[ ] 𝑎𝑑−𝑏𝑐 −𝑐 𝑎 1 𝑎 𝑏  (D) ⋅[ ] 𝑎𝑑−𝑏𝑐 𝑐 𝑑 Answer: 1. (C) Here are 5 MCQs based on the additional properties of invertible matrices: 1. If matrix 𝐴 is symmetric and invertible, what can be said about 𝐴−1 ?   (A) 𝐴−1 is diagonal  (B) 𝐴−1 is symmetric  (C) 𝐴−1 is triangular  (D) 𝐴−1 is skew-symmetric 2. If 𝐴 is a diagonal matrix and ∣ 𝐴 ∣≠ 0, which of the following is true?  (A) 𝐴−1 is triangular  (B) 𝐴−1 is skew-symmetric  (C) 𝐴−1 is diagonal  (D) 𝐴−1 is symmetric 3. For a scalar matrix 𝐴 = 𝑐𝐼𝑛 , where 𝑐 ≠ 0, what can be said about 𝐴−1 ?  (A) 𝐴−1 is triangular  (B) 𝐴−1 is also a scalar matrix  (C) 𝐴−1 is skew-symmetric  (D) 𝐴−1 is diagonal but not scalar 4. If 𝐴 is an upper triangular matrix and ∣ 𝐴 ∣≠ 0, what can be said about 𝐴−1 ?  (A) 𝐴−1 is symmetric  (B) 𝐴−1 is diagonal  (C) 𝐴−1 is also upper triangular  (D) 𝐴−1 is skew-symmetric 5. Which of the following is always true for an invertible matrix?  (A) It has more than one inverse  (B) It possesses a unique inverse  (C) It has infinitely many inverses  (D) The inverse must be symmetric Answers: 1. (B) 2. (C) 3. (B) 4. (C) 5. (B) Here are 7 MCQs based on the given information about matrix products and their determinants: 1. If 𝐴𝐵 = 𝑂, where 𝐴 and 𝐵 are square matrices, which of the following is true?  (A) 𝐴 must be the zero matrix  (B) At least one of 𝐴 or 𝐵 must be singular  (C) 𝐵 must be invertible  (D) Both 𝐴 and 𝐵 must be invertible 2. If the product of two matrices 𝐴 and 𝐵 is the zero matrix (𝐴𝐵 = 𝑂), and neither matrix is the zero matrix, what can be said about their determinants?  (A) Both ∣ 𝐴 ∣ and ∣ 𝐵 ∣ are non-zero  (B) Both ∣ 𝐴 ∣ and ∣ 𝐵 ∣ must be zero  (C) Only ∣ 𝐴 ∣ is zero  (D) Only ∣ 𝐵 ∣ is zero 3. If 𝐴𝐵 = 𝑂 and matrix 𝐴 is non-singular, what must be true?  (A) 𝐵 must be the zero matrix  (B) 𝐴 must be singular  (C) 𝐵 is invertible  (D) Neither 𝐴 nor 𝐵 is singular 4. If 𝐴𝐵 = 𝑂 and matrix 𝐵 is singular, what can we conclude?  (A) 𝐴 is invertible  (B) 𝐴 may be singular  (C) 𝐴 must be the identity matrix  (D) 𝐵 is invertible 5. If 𝐴𝐵 = 𝑂 and 𝐴 is the zero matrix, what can be said about 𝐵?  (A) 𝐵 must be singular   (B) 𝐵 can be any matrix  (C) 𝐵 must be invertible  (D) 𝐵 must also be the zero matrix 6. If the determinant of matrix 𝐴 is zero and 𝐴𝐵 = 𝑂, which of the following is necessarily true?  (A) 𝐵 must be invertible  (B) ∣ 𝐵 ∣≠ 0  (C) 𝐴 is singular  (D) 𝐵 must be the zero matrix 7. If 𝐴𝐵 = 𝑂 and both matrices 𝐴 and 𝐵 are non-singular, what is the result?  (A) The product 𝐴𝐵 cannot be zero  (B) It contradicts the assumption since 𝐴𝐵 = 𝑂  (C) Both matrices must be diagonal  (D) 𝐴 and 𝐵 must be identity matrices Answers: 1. (B) 2. (B) 3. (A) 4. (B) 5. (B) 6. (C) 7. (B) Here are 5 MCQs based on the Cayley-Hamilton Theorem and the characteristic equation for 2x2 matrices: 1. What does the Cayley-Hamilton Theorem state about every square matrix?  (A) It has a unique inverse  (B) It satisfies its own characteristic equation  (C) It can be diagonalized  (D) Its determinant is always non-zero 2. For a 2x2 matrix 𝐴, which of the following represents the characteristic equation?  (A) 𝐴2 − tr(𝐴) ⋅ 𝐴+∣ 𝐴 ∣⋅ 𝐼 = 0  (B) 𝐴2 + tr(𝐴) ⋅ 𝐴+∣ 𝐴 ∣⋅ 𝐼 = 0  (C) 𝐴2 +∣ 𝐴 ∣⋅ 𝐴 + tr(𝐴) ⋅ 𝐼 = 0  (D) 𝐴2 −∣ 𝐴 ∣⋅ 𝐴 + tr(𝐴) ⋅ 𝐼 = 0 3. In the characteristic equation ∣ 𝐴 − 𝜆𝐼 ∣= 0, what does 𝜆 represent?  (A) The determinant of 𝐴  (B) A scalar variable  (C) The trace of 𝐴  (D) The identity matrix 𝑎 𝑏 4. What is the trace of a 2x2 matrix 𝐴 = [ ]? 𝑐 𝑑   (A) 𝑎𝑑 − 𝑏𝑐  (B) 𝑎 + 𝑑  (C) 𝑎 ⋅ 𝑑  (D) 𝑏 + 𝑐 5. If a 2x2 matrix 𝐴 has a trace of 5 and a determinant of 6, what is the characteristic equation?  (A) 𝐴2 − 5𝐴 + 6𝐼 = 0  (B) 𝐴2 + 5𝐴 + 6𝐼 = 0  (C) 𝐴2 − 6𝐴 + 5𝐼 = 0  (D) 𝐴2 + 6𝐴 + 5𝐼 = 0 Answers: 1. (B) 2. (A) 3. (B) 4. (B) 5. (A) Here are 3 MCQs based on the shortcut for finding the characteristic equation of a 3x3 matrix: 1. In the characteristic equation for a 3x3 matrix 𝐴, what does the term 𝑡 represent?  (A) The determinant of 𝐴  (B) The inverse of 𝐴   (C) The trace of 𝐴  (D) The adjoint of 𝐴 2. Which of the following is the correct form of the characteristic equation for a 3x3 matrix 𝐴?  (A) 𝐴3 + 𝑡𝐴2 + 𝑘𝐴 + 𝑑𝐼 = 0  (B) 𝐴3 − 𝑡𝐴2 − 𝑘𝐴 − 𝑑𝐼 = 0  (C) 𝐴3 − 𝑡𝐴2 + 𝑘𝐴 − 𝑑𝐼 = 0  (D) 𝐴3 + 𝑡𝐴2 − 𝑘𝐴 + 𝑑𝐼 = 0 3. What is the value of 𝑘 in the characteristic equation of a 3x3 matrix 𝐴?  (A) tr(𝐴)  (B) ∣ 𝐴 ∣ 1 2  (C) ((tr(𝐴)) − tr(𝐴2 )) 2  (D) tr(𝐴2 )−∣ 𝐴 ∣ Answers: 1. (C) 2. (C) 3. (C) Here are 10 MCQs based on Cramer's Rule for solving systems of linear equations: 1. What is the first step in Cramer's Rule for solving a system of linear equations?  (A) Calculate Δx  (B) Calculate Δ (the determinant of the coefficient matrix)  (C) Check if the system has infinitely many solutions  (D) Find the inverse of the coefficient matrix 2. When using Cramer's Rule, what does it indicate if Δ ≠ 0?  (A) The system has no solution  (B) The system has infinitely many solutions  (C) The system is consistent and has a unique solution  (D) The system is inconsistent and has a unique solution 3. In Cramer's Rule, what should you check if Δ = 0?  (A) The value of the inverse matrix  (B) The values of x, y, and z directly  (C) The determinants Δx, Δy, and Δz  (D) The number of equations in the system 4. If Δ = 0 and at least one of Δx, Δy, or Δz is not zero, what can be concluded?  (A) The system has a unique solution   (B) The system is consistent  (C) The system is inconsistent (no solution)  (D) The system has infinitely many solutions 5. When all of Δx, Δy, and Δz are zero, what type of solution does the system have?  (A) A unique solution  (B) Infinitely many solutions  (C) No solution  (D) A contradictory solution 6. What does Cramer's Rule require about the number of equations and unknowns?  (A) More equations than unknowns  (B) Fewer equations than unknowns  (C) The same number of equations as unknowns  (D) At least one equation must be linear 7. In Cramer's Rule, which variable can be expressed in terms of a parameter when the system has infinitely many solutions?  (A) z  (B) x  (C) y  (D) Any variable 8. What is the significance of the determinant Δ in Cramer's Rule?  (A) It determines the nature of the solutions of the system  (B) It represents the number of variables in the system  (C) It calculates the value of the variables directly  (D) It provides the inverse of the coefficient matrix 9. Which of the following scenarios is possible when Δ = 0?  (A) The system has a unique solution  (B) The system can either be inconsistent or have infinitely many solutions  (C) The system is guaranteed to have a unique solution  (D) The determinant values are irrelevant 10. If you find Δ ≠ 0 in Cramer's Rule, what is the next step?  (A) Find Δx, Δy, and Δz  (B) Solve for the variables directly using the determinant  (C) Check for inconsistencies in the equations  (D) Determine the eigenvalues of the matrix Answers: 1. (B) 2. (C) 3. (C) 4. (C) 5. (B) 6. (C) 7. (A) 8. (A) 9. (B) 10. (B)