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.
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 ›
Q2.
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 ›
Q3.
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 ›
Q4.
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 ›
Q5.
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 ›
Q6.
Consider the matrix $B = (- 1 - 120 - 1 - 100 - 1)$ . The sum of all the entries of the matrix $B^{19}$ is
A. $- 163$ B. $- 171$ C. $- 192$ D. $- 174$
hardAsked in NIMCET 2025View solution ›
Q7.
Let $A$ and $B$ be two square matrices of the same order satisfying $A^{2} + 5 A + 5 I = O$ and $B^{2} + 3 B + I = O$ respectively, where $I$ is the identity matrix. Then the inverse of the matrix $C = B A + 2 B + 2 A + 4 I$ is
A. $B A - 3 B + A - 3 I$ B. $A B + A + 3 B + 3 I$ C. $B A + 3 B + A + 3 I$ D. $A B - A + 3 B - 3 I$
hardAsked in NIMCET 2025View solution ›