Questions and Answers
Which of the following matrices can be added to a $2 \times 3$ matrix?
Given matrices $A$ of order $3 \times 2$ and $B$ of order $2 \times 3$, what is the size of the product matrix $AB$?
Which of the following is true for two matrices $A$ and $B$ such that $A \times B$ is defined?
What property is verified by $A + B = B + A$ for matrices $A$ and $B$?
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Which type of matrix has only one row?
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What is the additive identity matrix for any $m \times n$ matrix?
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When is the product of two matrices $A$ and $B$ equal to the product of $B$ and $A$ ($AB = BA$)?
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Given a matrix $A$ is symmetric if and only if:
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Which of the following expressions represents the distributive property of matrices?
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The multiplicative inverse of a matrix $A$ is defined such that:
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Study Notes
Matrices
- A rectangular array of real numbers enclosed within brackets is said to form a matrix.
- A matrix can be classified into different types based on its order:
- Rectangular matrix: number of rows and columns are not equal.
- Square matrix: number of rows is equal to the number of columns.
- Row matrix: has only one row.
- Column matrix: has only one column.
- Null/zero matrix: each of its entries is 0.
Operations on Matrices
- Transpose of a matrix: obtained by interchanging rows into columns or columns into rows.
- Symmetric matrix: a square matrix that is equal to its transpose (A = A').
- Negative of a matrix: obtained by changing the signs of the entries of the matrix (-A).
- Addition of matrices: can be done if they have the same order, by adding corresponding entries.
- Multiplication of matrices: can be done if the number of columns in the first matrix is equal to the number of rows in the second matrix.
Square Matrices
- Diagonal: a line of elements from the top left to the bottom right of a square matrix.
- Diagonal matrix: a square matrix with all elements zero except for the elements on the main diagonal.
- Identity matrix: a diagonal matrix with all diagonal entries equal to 1.
- Determinant of a square matrix: a scalar value denoted by det(M) or |M|.
- Singular matrix: a square matrix with a determinant equal to zero.
- Non-singular matrix: a square matrix with a determinant not equal to zero.
- Adjoint of a matrix: defined for a 2x2 matrix.
- Inverse of a matrix: defined for a non-singular square matrix.
Laws of Matrices
- Commutative law of addition: M + N = N + M.
- Associative law of addition: (M + N) + T = M + (N + T).
- Distributive laws: M(N+T) = MN + MT, (M+N)T = MT + NT.
- Law of transpose of product: (AB)' = B'A'.
Applications of Matrices
- Solution of simultaneous linear equations: can be expressed in matrix form and solved using matrices.
- Cramer's rule: a method for solving simultaneous linear equations using determinants.
Real Numbers
- Properties of real numbers: recall the set of real numbers, depict on a number line, demonstrate operations on the number line.
- Radicals and laws of indices: recall basic properties, apply the laws of exponents and logarithms.
- Complex numbers: define, recognize, and manipulate complex numbers, define conjugate, and apply properties.
Review Exercises
- Answer multiple-choice questions on matrices, including order, types, and operations.
- Complete exercises on matrix multiplication, addition, and transpose.
- Solve problems involving matrices and determinants.
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Description
Learn about different types of matrices in linear algebra, including rectangular, square, row, column, and null matrices.
