Class 12 • mathematics • Chapter 4
DETERMINANTS
True & False Quiz
Expand. Evaluate. Invert.
✓True
✗False
25
Questions
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Ch.4
Chapter
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XII
Class
Why True & False for DETERMINANTS?
How this format sharpens your conceptual clarity
🔵 The determinant is a single number that decides invertibility and gives area/volume interpretations for matrices.
✅ T/F tests minors vs cofactors, the adjoint-inverse formula, and when a system of equations has a unique solution.
🎯 Trap: a square matrix is invertible if and only if its determinant is NON-ZERO — |A| = 0 means A is singular.
📋
Read each statement carefully. Click True or False — instant feedback with explanation appears. Submit anytime; unattempted questions are marked Skipped.
Q 1
For a square matrix of order \(2\), \(\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc\).
Q 2
If two rows of a determinant are identical, its value is zero.
Q 3
The determinant of the identity matrix \(I_n\) is \(0\) for every positive integer \(n\).
Q 4
If every element of one row of a determinant is multiplied by \(k\), the value of the determinant is multiplied by \(k\).
Q 5
If every element of a \(3\times3\) determinant is multiplied by \(2\), its value becomes twice the original value.
Q 6
If one row of a determinant is entirely zero, the determinant is zero.
Q 7
Interchanging any two rows of a determinant leaves its numerical value unchanged.
Q 8
For any square matrix \(A\), \(\det(A^T)=\det(A)\).
Q 9
If the determinant of a square matrix is zero, the matrix must be the identity matrix.
Q 10
If one row of a determinant is replaced by itself plus a multiple of another row, the determinant remains unchanged.
Q 11
For a \(3\times3\) matrix \(A\), \(\det(2A)=2\det(A)\).
Q 12
If \(A\) is a square matrix and \(\det(A)\neq0\), then \(A\) is invertible.
Q 13
The determinant of a triangular matrix is equal to the product of its diagonal elements.
Q 14
If \(A\) and \(B\) are square matrices of the same order, then \(\det(A+B)=\det(A)+\det(B)\).
Q 15
If two rows of a determinant are proportional, then the determinant is zero.
Q 16
If \(\det(A)=5\) for a \(3\times3\) matrix \(A\), then \(\det(3A)=45\).
Q 17
If \(\det(A)=-4\), then \(\det(A^T)=-4\).
Q 18
If \(A\) is a \(3\times3\) matrix with \(\det(A)=0\), then \(\det(5A)=0\).
Q 19
For square matrices \(A\) and \(B\) of the same order, \(\det(AB)=\det(A)\det(B)\).
Q 20
If \(\det(A)=2\) and \(\det(B)=-3\), then \(\det(AB)=1\).
Q 21
If \(A\) is an invertible square matrix, then \(\det(A^{-1})=\dfrac{1}{\det(A)}\).
Q 22
If \(A\) is a \(3\times3\) matrix with \(\det(A)=2\), then \(\det(\operatorname{adj}A)=4\).
Q 23
If \(A\) is a \(3\times3\) matrix and \(\det(A)=-2\), then \(\det(\operatorname{adj}A)=-4\).
Q 24
If \(A\) is a nonsingular \(3\times3\) matrix, then \(A^{-1}=\dfrac{\operatorname{adj}A}{\det(A)}\).
Q 25
For a \(3\times3\) matrix \(A\), if \(\det(A)=0\), then the system \(AX=B\) has a unique solution for every \(B\).
Key Takeaways — DETERMINANTS
Core facts for CBSE Boards & JEE
1
A square matrix A is invertible ⇔ |A| ≠ 0; if |A| = 0, A is called singular.
2
Cofactor Cᵢⱼ = (−1)⁺ᵀ × Minor Mᵢⱼ — the sign alternates by position.
3
A⁻¹ = (1/|A|) × adj(A), valid only when |A| ≠ 0.
4
Area of a triangle with given vertices uses a 3×3 determinant; a zero result means the points are collinear.
5
|AB| = |A||B| for square matrices of the same order (determinant of a product).
6
Interchanging any two rows (or columns) of a determinant changes its sign but not its magnitude.