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Questions and Answers
What is the necessary and sufficient condition for a square matrix A to be invertible?
What is the necessary and sufficient condition for a square matrix A to be invertible?
- A is an identity matrix
- A is a square matrix of any order
- A is a nonsingular matrix (correct)
- A is a diagonal matrix
If A and B are nonsingular matrices of the same order, which of the following statements is true?
If A and B are nonsingular matrices of the same order, which of the following statements is true?
- AB is nonsingular, but BA is singular
- AB and BA are both nonsingular matrices (correct)
- AB is singular, but BA is nonsingular
- AB and BA are both singular matrices
If A and B are square matrices of the same order, what is the relationship between the determinants of A, B, and their product AB?
If A and B are square matrices of the same order, what is the relationship between the determinants of A, B, and their product AB?
- |AB| = |A| + |B|
- |AB| = |A| - |B|
- |AB| = |A| * |B| (correct)
- |AB| = |A| / |B|
What is the value of the determinant of the adjoint of a square matrix A of order n?
What is the value of the determinant of the adjoint of a square matrix A of order n?
If A is an invertible matrix, which of the following statements is true?
If A is an invertible matrix, which of the following statements is true?
If A is a $3 \times 3$ matrix and $|A| = 5$, what is the value of $|adj(A)|$?
If A is a $3 \times 3$ matrix and $|A| = 5$, what is the value of $|adj(A)|$?
If A is a nonsingular matrix, what is the inverse of A, denoted by A^(-1)?
If A is a nonsingular matrix, what is the inverse of A, denoted by A^(-1)?
If A and B are nonsingular matrices of the same order, which of the following statements is true about their inverses?
If A and B are nonsingular matrices of the same order, which of the following statements is true about their inverses?
If A is a nonsingular matrix and I is the identity matrix of the same order, which of the following statements is true?
If A is a nonsingular matrix and I is the identity matrix of the same order, which of the following statements is true?
If A is a $2 \times 2$ matrix with $|A| = 6$, what is the value of $|adj(A)|$?
If A is a $2 \times 2$ matrix with $|A| = 6$, what is the value of $|adj(A)|$?
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Study Notes
Nonsingular Matrices
- A matrix A is nonsingular if its determinant is not equal to zero.
- If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.
Determinant of Product of Matrices
- The determinant of the product of matrices is equal to the product of their respective determinants.
- If A is a square matrix of order n, then |adj(A)| = |A|^(n-1).
Invertible Matrices
- A square matrix A is invertible if and only if A is a nonsingular matrix.
- If A is invertible, then A^(–1) = (1/|A|)adj(A).
Solution of System of Linear Equations
- The system of linear equations can be written as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
- If A is a nonsingular matrix, then the system of equations has a unique solution given by X = A^(–1)B.
- If A is a singular matrix, then the system of equations may have no solution or infinitely many solutions.
Adjoint of a Matrix
- The adjoint of a square matrix A is defined as the transpose of the matrix of cofactors.
- Adjoint of the matrix A is denoted by adj(A).
- adj(A) can be used to find the inverse of a matrix A, i.e., A^(–1) = (1/|A|)adj(A).
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