Properties of Determinants — JEE Main Maths PYQ MCQs with Solutions
Free JEE Main Maths PYQ Properties of Determinants MCQs with step-by-step solutions (20 questions). Part of Matrices & Determinants. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Properties of Determinants · easy · theory
If $A$ is a square matrix of order 3, then $\det(2A)$ equals:
A. $2\det(A)$
B. $4\det(A)$
C. $8\det(A)$ ✓ Correct
D. $16\det(A)$
Solution: For a square matrix $A$ of order $n$, $\det(kA) = k^n\det(A)$. Here $n=3$ and $k=2$, so $\det(2A) = 2^3\det(A) = 8\det(A)$.
Q2 — Properties of Determinants · easy · theory
If $\det(A) = 5$ and $\det(B) = 3$, where $A$ and $B$ are square matrices of the same order, then $\det(AB)$ is:
A. $2$
B. $8$
C. $15$ ✓ Correct
D. $5/3$
Solution: By the multiplicative property of determinants: $\det(AB) = \det(A) \cdot \det(B) = 5 \times 3 = 15$.
Q3 — Properties of Determinants · easy · theory
If swapping two rows of a matrix changes the determinant from $D$ to $-D$, which property of determinants is being illustrated?
A. Row scaling property
B. Row addition property
C. Row swap property ✓ Correct
D. Transposition property
Solution: Swapping two rows (or columns) of a matrix multiplies the determinant by $-1$. This is a fundamental property used in row reduction and Gaussian elimination.
Q4 — Properties of Determinants · easy · theory
If matrix $A$ has a row of all zeros, then $\det(A)$ is:
A. $1$
B. $0$ ✓ Correct
C. $-1$
D. Undefined
Solution: If a matrix has a row (or column) of all zeros, the determinant is always $0$. This is because the rows (columns) are linearly dependent.
Q5 — Properties of Determinants · easy · theory
If two rows of a matrix are identical, then $\det(A)$ is:
A. $1$
B. $0$ ✓ Correct
C. $-1$
D. Positive
Solution: If two rows (or columns) of a matrix are identical, the rows are linearly dependent, making the determinant equal to $0$.
Q6 — Properties of Determinants · easy · theory
If $\det(A) = 7$, then $\det(A^T)$ is:
A. $-7$
B. $1/7$
C. $7$ ✓ Correct
D. $0$
Solution: The determinant of a transpose equals the determinant of the original matrix: $\det(A^T) = \det(A)$. This is because row operations on $A$ correspond to column operations on $A^T$.
Q7 — Properties of Determinants · medium · theory
For a symmetric matrix $A$ (where $A = A^T$), if $\det(A) = -3$, what is the nature of the matrix?
A. Singular
B. Non-singular ✓ Correct
C. Idempotent
D. Nilpotent
Solution: A matrix is singular if $\det(A) = 0$ and non-singular if $\det(A) \neq 0$. Since $\det(A) = -3 \neq 0$, the matrix $A$ is non-singular. Symmetry does not affect singularity.
Q8 — Properties of Determinants · medium · theory
If $A$ is a skew-symmetric matrix of order $3$ (where $A^T = -A$), then $\det(A)$ is always:
A. Positive
B. Negative
C. $0$ ✓ Correct
D. $1$
Solution: For a skew-symmetric matrix $A$ of odd order, $\det(A) = 0$. Proof: $\det(A^T) = \det(-A)$, so $\det(A) = (-1)^3\det(A) = -\det(A)$, which implies $\det(A) = 0$.
Q9 — Properties of Determinants · medium · theory
Adding $k$ times row $i$ to row $j$ in a matrix $A$ (where $i \neq j$):
A. Multiplies $\det(A)$ by $k$
B. Multiplies $\det(A)$ by $-k$
C. Does not change $\det(A)$ ✓ Correct
D. Multiplies $\det(A)$ by $1/k$
Solution: Row addition (adding a multiple of one row to another) is an elementary row operation that does not change the determinant. This property is crucial for computing determinants using row reduction.
Q10 — Properties of Determinants · medium · theory
If $A$ is an orthogonal matrix (where $AA^T = A^T A = I$), then $\det(A)$ is:
A. $\pm 1$ ✓ Correct
B. $0$
C. Any real number
D. $-1$ always
Solution: For an orthogonal matrix $A$, we have $\det(AA^T) = \det(I)$. Thus $\det(A)\det(A^T) = 1$, and since $\det(A^T) = \det(A)$, we get $[\det(A)]^2 = 1$, so $\det(A) = \pm 1$.
Q11 — Properties of Determinants · medium · theory
For matrices $A$ and $B$ of the same order, which of the following is generally true?
A. $\det(A+B) = \det(A) + \det(B)$
B. $\det(A-B) = \det(A) - \det(B)$
C. $\det(AB) = \det(A)\det(B)$ ✓ Correct
D. $\det(A+B) = \det(A)\det(B)$
Solution: The only property that holds in general is $\det(AB) = \det(A)\det(B)$ (multiplicative property). Determinant is NOT additive; $\det(A+B) \neq \det(A) + \det(B)$ in general.
Q12 — Properties of Determinants · medium · theory
If a matrix $A$ is obtained from matrix $B$ by multiplying a row by the scalar $c$, and $\det(B) = D$, then $\det(A)$ is:
A. $D$
B. $cD$ ✓ Correct
C. $D/c$
D. $c^2 D$
Solution: Multiplying a single row (or column) of a matrix by a scalar $c$ multiplies the determinant by $c$. Thus $\det(A) = c \cdot \det(B) = cD$.
Q13 — Properties of Determinants · medium · theory
If the determinant of a $3 \times 3$ matrix is $24$, and all elements are doubled, the new determinant is:
A. $24$
B. $48$
C. $96$
D. $192$ ✓ Correct
Solution: When all elements of a $3 \times 3$ matrix are multiplied by $2$, the determinant is multiplied by $2^3 = 8$. New determinant $= 8 \times 24 = 192$.
Q14 — Properties of Determinants · easy · theory
For an upper triangular matrix, the determinant equals:
A. Sum of diagonal elements
B. Product of diagonal elements ✓ Correct
C. Sum of all elements
D. Zero always
Solution: For any triangular matrix (upper or lower), the determinant is the product of all diagonal elements. This makes computing determinants easy for triangular matrices obtained via row reduction.
Q15 — Properties of Determinants · easy · theory
If $\det(A) = 0$, then $A$ is:
A. Invertible
B. Singular (non-invertible) ✓ Correct
C. Orthogonal
D. Symmetric
Solution: A square matrix is invertible if and only if $\det(A) \neq 0$. If $\det(A) = 0$, the matrix is singular and does not have an inverse.
Q16 — Properties of Determinants · medium · theory
If $A^{-1}$ exists and $\det(A) = 5$, then $\det(A^{-1})$ is:
A. $5$
B. $1/5$ ✓ Correct
C. $-5$
D. $-1/5$
Solution: Since $AA^{-1} = I$, we have $\det(A)\det(A^{-1}) = \det(I) = 1$. Thus $\det(A^{-1}) = 1/\det(A) = 1/5$.
Q17 — Properties of Determinants · medium · theory
Reducing the determinant $\begin{vmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{vmatrix}$ using row operations gives:
A. $1$
B. $6$
C. $0$ ✓ Correct
D. $9$
Solution: Notice that row 2 is $2 \times$ row 1 and row 3 is $3 \times$ row 1. The rows are linearly dependent. Using row operations $R_2 - 2R_1 \to R_2$ and $R_3 - 3R_1 \to R_3$ produces two rows of zeros, making $\det(A) = 0$.
Q18 — Properties of Determinants · medium · theory
If $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = k$, then $\det\begin{pmatrix} c & d \\ a & b \end{pmatrix}$ equals:
A. $k$
B. $-k$ ✓ Correct
C. $k^2$
D. $0$
Solution: Swapping rows 1 and 2 changes the sign of the determinant. The original has $\det(A) = ad - bc = k$. After swapping rows: $\det\begin{pmatrix} c & d \\ a & b \end{pmatrix} = cb - da = -(ad - bc) = -k$.
Q19 — Properties of Determinants · easy · theory
For a diagonal matrix $D = \begin{pmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{pmatrix}$, $\det(D)$ equals:
A. $d_1 + d_2 + d_3$
B. $d_1 \cdot d_2 \cdot d_3$ ✓ Correct
C. $d_1 \cdot d_2 + d_2 \cdot d_3 + d_3 \cdot d_1$
D. $0$
Solution: A diagonal matrix is a special case of a triangular matrix. The determinant of a diagonal matrix is the product of its diagonal elements: $\det(D) = d_1 \cdot d_2 \cdot d_3$.
Q20 — Properties of Determinants · hard · theory
If $A$ is a matrix such that $A^2 = A$ (idempotent), and $\det(A) \neq 0$, then $A$ must be:
A. A zero matrix
B. The identity matrix ✓ Correct
C. An orthogonal matrix
D. A symmetric matrix
Solution: If $A^2 = A$, then $A^2 - A = 0$, so $A(A-I) = 0$. Taking determinants: $\det(A)\det(A-I) = 0$. Since $\det(A) \neq 0$, we must have $\det(A-I) = 0$. For an idempotent matrix with non-zero determinant, $A = I$, which gives $\det(I) = 1$.