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Ch-3: Matrices
12 Questions
Before You Practice Try each question on your own first. Use the Show Answer button only after you have selected your option, then read the explanation and revise the related concept from your chapter notes.
1. A matrix in which the number of rows equals the number of columns is called a:
A. Square matrix
B. Row matrix
C. Column matrix
D. Diagonal matrix
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Answer: A. Square matrix
Explanation: A matrix with an equal number of rows and columns (order n × n) is called a square matrix.
2. Two matrices can be multiplied only when:
A. The number of columns of the first equals the number of rows of the second
B. Both matrices have the same order
C. Both matrices are square
D. Both matrices are symmetric
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Answer: A. The number of columns of the first equals the number of rows of the second
Explanation: Matrix multiplication A·B is defined only when the column count of A matches the row count of B.
3. A square matrix A is called symmetric if:
A. A = A' (its transpose)
B. A = -A'
C. A² = A
D. A · A' = I
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Answer: A. A = A' (its transpose)
Explanation: A symmetric matrix is one that is equal to its own transpose.
4. A square matrix A is called skew-symmetric if:
A. A' = -A
B. A' = A
C. A² = I
D. A is diagonal
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Answer: A. A' = -A
Explanation: A skew-symmetric matrix satisfies A' = -A, which also forces all diagonal elements to be zero.
5. The identity matrix I acts as the:
A. Multiplicative identity for matrix multiplication
B. Additive identity
C. Zero element of the matrix set
D. Inverse of every matrix
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Answer: A. Multiplicative identity for matrix multiplication
Explanation: For any square matrix A, A·I = I·A = A, similar to how 1 behaves in ordinary multiplication.
6. For any square matrix A, the matrix (A + A') is always:
A. Symmetric
B. Skew-symmetric
C. A zero matrix
D. An identity matrix
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Answer: A. Symmetric
Explanation: Taking the transpose of (A + A') gives (A' + A), which equals the original expression, confirming it is symmetric.
7. For any square matrix A, the matrix (A - A') is always:
A. Skew-symmetric
B. Symmetric
C. An identity matrix
D. A diagonal matrix
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Answer: A. Skew-symmetric
Explanation: Taking the transpose of (A - A') gives (A' - A) = -(A - A'), confirming it is skew-symmetric.
8. A matrix in which every element is zero is called a:
A. Null (zero) matrix
B. Identity matrix
C. Scalar matrix
D. Diagonal matrix
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Answer: A. Null (zero) matrix
Explanation: A matrix with all entries equal to zero is called the null or zero matrix, regardless of its order.
9. Matrix multiplication is, in general:
A. Not commutative
B. Always commutative
C. The same operation as matrix addition
D. Associative only for square matrices
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Answer: A. Not commutative
Explanation: In general, AB ≠ BA for matrices, unlike multiplication of real numbers.
10. The order of a matrix having 3 rows and 2 columns is written as:
A. 3 × 2
B. 2 × 3
C. 3 × 3
D. 2 × 2
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Answer: A. 3 × 2
Explanation: By convention, the order of a matrix is expressed as (number of rows) × (number of columns).
11. A square matrix A is invertible if and only if:
A. |A| ≠ 0
B. |A| = 0
C. A is symmetric
D. A is a diagonal matrix
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Answer: A. |A| ≠ 0
Explanation: A square matrix has an inverse precisely when its determinant is non-zero (i.e., it is non-singular).
12. If A and B are matrices of the same order, then (A + B)' equals:
A. A' + B'
B. A' - B'
C. B' - A'
D. AB'
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Answer: A. A' + B'
Explanation: The transpose of a sum of matrices equals the sum of their individual transposes.
