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Ch-4: Determinants
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. The determinant of the 2×2 matrix [[a, b], [c, d]] is:
A. ad - bc
B. ac - bd
C. ad + bc
D. ab - cd
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Answer: A. ad - bc
Explanation: For a 2×2 matrix, the determinant is calculated as the product of the main diagonal minus the product of the other diagonal.
2. If the determinant of a square matrix is zero, the matrix is called:
A. Singular
B. Non-singular
C. Orthogonal
D. Symmetric
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Answer: A. Singular
Explanation: A matrix with a zero determinant is termed singular and does not have an inverse.
3. The area of a triangle with given coordinate vertices can be conveniently computed using:
A. Determinants
B. Only the distance formula
C. Only the slope formula
D. Matrix addition
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Answer: A. Determinants
Explanation: The area of a triangle formed by three points can be expressed compactly as half the absolute value of a 3×3 determinant.
4. If two rows (or columns) of a determinant are identical, the value of the determinant is:
A. Zero
B. One
C. The sum of all elements
D. Undefined
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Answer: A. Zero
Explanation: Identical rows or columns make the determinant vanish, since one row can be obtained from the other by a scalar multiple of 1.
5. Cramer's rule is primarily used to solve:
A. A system of linear equations
B. Quadratic equations
C. Differential equations
D. Definite integrals
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Answer: A. A system of linear equations
Explanation: Cramer's rule uses determinants to directly express the solution of a system of linear equations with a unique solution.
6. The adjoint of a square matrix A is defined as the:
A. Transpose of the cofactor matrix of A
B. Same as the inverse of A
C. Determinant of A
D. Matrix A itself
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Answer: A. Transpose of the cofactor matrix of A
Explanation: The adjoint (adjugate) is obtained by taking the transpose of the matrix formed from the cofactors of A.
7. The fundamental relation between a square matrix A, its adjoint, and its determinant is:
A. A · (adj A) = |A| · I
B. A · (adj A) = A
C. adj A = A⁻¹ always
D. |A| = adj A
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Answer: A. A · (adj A) = |A| · I
Explanation: This identity is the basis for deriving the formula for the inverse of a matrix using its adjoint.
8. If |A| ≠ 0 for a square matrix A, its inverse is given by:
A. A⁻¹ = (1/|A|) · adj A
B. A⁻¹ = |A| · adj A
C. A⁻¹ = adj A alone
D. A⁻¹ = A
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Answer: A. A⁻¹ = (1/|A|) · adj A
Explanation: The inverse formula follows directly from the identity A · (adj A) = |A| · I.
9. Interchanging any two rows (or columns) of a determinant:
A. Changes its sign
B. Leaves it unchanged
C. Makes it zero
D. Doubles its value
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Answer: A. Changes its sign
Explanation: Row or column interchange is one of the fundamental properties of determinants that reverses the sign of the value.
10. The determinant of an identity matrix, regardless of its order, is:
A. 1
B. 0
C. Equal to its order n
D. -1
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Answer: A. 1
Explanation: Since all off-diagonal elements are zero and all diagonal elements are 1, the determinant of I is always 1.
11. If a determinant has a row (or column) consisting entirely of zeros, its value is:
A. Zero
B. One
C. Undefined
D. Equal to the trace of the matrix
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Answer: A. Zero
Explanation: Expanding along a row or column of all zeros yields a determinant value of zero.
12. A system of linear equations has a unique solution when the coefficient determinant is:
A. Non-zero
B. Zero
C. Negative
D. Always equal to one
Show Answer
Answer: A. Non-zero
Explanation: By Cramer's rule, a non-zero coefficient determinant guarantees a unique solution to the system.
