Matrices Questions with Solutions

25 questions7 easy11 medium7 hard

A matrix is a rectangular arrangement of numbers in rows and columns. This topic covers types of matrices (square, diagonal, scalar, identity, symmetric and skew-symmetric), matrix operations (addition, scalar multiplication and matrix multiplication), the transpose, determinants and their properties, minors and cofactors, the adjoint, and the inverse of a matrix. It also covers solving systems of linear equations using the inverse method and elementary row and column transformations. Remember that matrix multiplication is not commutative, so AB need not equal BA, and that the inverse exists only when the determinant is not zero. Useful results include |AB| = |A||B|, (AB)^-1 = B^-1 A^-1, and |kA| = k^n |A| for a matrix of order n. These properties solve many questions without heavy calculation. Practise the matrices questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
The value of the determinant of the following matrix at $x = 2026$ is: $∣ xx + 1 x + 3 x + 1 x + 3 x + 6 x + 3 x + 6 x + 10 ∣$
A. −1B. 0C. 1D. 2026
mediumAsked in NIMCET 2026View solution ›
Q2.
The number of values of $θ$ in the interval $[0, 2 π]$ for which the following homogeneous system of equations has a non-trivial solution is: $x + (s in θ) y + (cos θ) z = 0$ $x + (cos θ) y + (s in θ) z = 0$ $x - (s in θ) y - (cos θ) z = 0$
A. 0B. 2C. 4D. 6
mediumAsked in NIMCET 2026View solution ›
Q3.
If $x, y, z$ satisfy the following equations simultaneously, then which of the following is true? $x + y + z = 1$ , $4 x + 9 y + 16 z = 25$ , $16 x + 81 y + 256 z = 625$
A. $x = - \frac{36}{15}$ B. $x = - \frac{15}{36}$ C. $x = \frac{15}{36}$ D. $x = \frac{36}{15}$
mediumAsked in NIMCET 2026View solution ›
Q4.
If $[α β γ - α]$ is to be square root of the two rowed unit matrix, then $α$ , $β$ and $γ$ should satisfy
A. $1 + α^{2} + β γ = 0$ B. $1 - α^{2} - β γ = 0$ C. $1 - α^{2} + β γ = 0$ D. $1 + α^{2} - β γ = 0$
mediumAsked in AMU MCA 2016View solution ›
Q5.
If A and B are square matrices such that $B = - A^{- 1} B A$ , then $(A + B)^{2}$ is
A. $O$ (the zero matrix)B. $A^{2} + B^{2}$ C. $A^{2} + 2 A B + B^{2}$ D. $A + B$
easyAsked in NIMCET 2023View solution ›
Q6.
If $A = [1000110 - 24]$ and $6 A^{- 1} = A^{2} + c A + d I$ , where $A^{- 1}$ is A-inverse and I is the identity matrix, then $(c, d)$ is
A. $(- 6, 11)$ B. $(6, - 11)$ C. $(11, - 6)$ D. $(6, 11)$
mediumAsked in NIMCET 2013View solution ›
Q7.
If $ω$ is a cube root of unity, then find the value of the determinant $∣ 1 + ω ω^{2} - ω 1 + ω^{2} ω - ω^{2} ω^{2} + ω ω - ω^{2} ∣$
A. $3 ω$ B. $- 3 ω$ C. $3 ω^{2}$ D. $- 3 ω^{2}$
hardAsked in NIMCET 2013View solution ›
Q8.
If $B = - A^{- 1} B A$ , then $(A + B)^{2}$ is equal to
A. $O$ B. $A^{2} + 2 A B + B^{2}$ C. $A^{2} + B^{2}$ D. $A + B$
mediumAsked in NIMCET 2011View solution ›
Q9.
The value of $a$ for which the system of equations $a^{3} x + (a + 1)^{3} y + (a + 2)^{3} z = 0$ $a x + (a + 1) y + (a + 2) z = 0$ $x + y + z = 0$ has a non-zero solution is
A. $1$ B. $0$ C. $- 1$ D. None of these
hardAsked in NIMCET 2010View solution ›
Q10.
If $a \neq = p$ , $b \neq = q$ , $c \neq = r$ and $∣ p b c a q c ab r ∣= 0$ , then the value of $\frac{p}{p - a} + \frac{q}{q - b} + \frac{r}{r - c}$ is
A. $0$ B. $1$ C. $- 1$ D. $2$
hardAsked in NIMCET 2010View solution ›
Q11.
If $A$ is a $3 \times 3$ matrix with $d e t (A) = 3$ , then $d e t (a d j A)$ is:
A. 3B. 9C. 27D. 6
easyAsked in NIMCET 2009View solution ›
Q12.
If $a + b + c \neq = 0$ , the system of equations $(b + c) (y + z) - a x = b - c$ $(c + a) (z + x) - b y = c - a$ $(a + b) (x + y) - cz = a - b$ has:
A. a unique solutionB. no solutionC. infinite number of solutionsD. finitely many solutions
hardAsked in NIMCET 2009View solution ›
Q13.
If $A = [cos α s in α - s in α cos α]$ , then for any positive integer n, $A^{n}$ is
A. $[s inn α cos n α cos n α - s inn α]$ B. $[cos n α s inn α s inn α cos n α]$ C. $[cos n α s inn α s inn α - cos n α]$ D. $[cos n α s inn α - s inn α cos n α]$
mediumAsked in NIMCET 2020View solution ›
Q14.
Let $A = (a_{ij})$ and $B = (b_{ij})$ be two square matrices of order n and det(A) denotes the determinant of A. Then, which of the following is not correct?
A. If A is a diagonal matrix, then $d e t (A) = a_{11} a_{22} \dots a_{nn}$ B. $d e t (A B) = d e t (A) d e t (B)$ C. $d e t (c A) = c d e t (A)$ D. $d e t (A) = d e t (A^{T})$ , where $A^{T}$ denotes the transpose of the matrix A
easyAsked in NIMCET 2020View solution ›
Q15.
If $F (θ) = [cos θ - s in θ 0 s in θ cos θ 0001]$ , then $F (θ) F (α)$ is equal to
A. $F (θ α)$ B. $F (\frac{θ}{α})$ C. $F (θ + α)$ D. $F (θ - α)$
easyAsked in NIMCET 2021View solution ›
Q16.
If the matrix $[- 1321 k - 3145]$ has an inverse matrix, then the value of k is:
A. k is any real numberB. $k \neq = - 4$ C. $k = - 4$ D. $k \neq = 4$
easyAsked in NIMCET 2014View solution ›
Q17.
If $A + B + C = π$ , then the value of $∣ s in (A + B + C) s in B cos C - s in B 0 t an A cos (A + B) - t an A 0 ∣$ is
A. 0B. 1C. $2 s in A s in B$ D. 2
mediumAsked in NIMCET 2015View solution ›
Q18.
A matrix $M_{r}$ is defined as $M_{r} = [r r - 1 r - 1 r]$ , $r \in N$ . Then the value of $d e t (M_{1}) + d e t (M_{2}) + \dots + d e t (M_{2015})$ is
A. $2014^{2}$ B. $2013^{2}$ C. 2014D. $2015^{2}$
easyAsked in NIMCET 2015View solution ›
Q19.
If $A = [0500]$ and $f (x) = 1 + x + x^{2} + \dots + x^{16}$ , then $f (A) =$
A. $0$ B. $[1501]$ C. $[1500]$ D. $[0511]$
easyAsked in NIMCET 2018View solution ›
Q20.
If a, b, c are the roots of the equation $x^{3} - 3 x^{2} + 3 x + 7 = 0$ , then the value of $∣ 2 b c - a^{2} c^{2} b^{2} c^{2} 2 a c - b^{2} a^{2} b^{2} a^{2} 2 ab - c^{2} ∣$ is
A. 9B. 27C. 81D. 0
hardAsked in NIMCET 2018View solution ›

Page 1 of 2Next ›