Class 12 Matrices Mock Test (CBSE)

Topics in Matrices

• Matrix and its order
• Types of matrices
• Equality of matrices
• Addition and scalar multiplication of matrices
• Multiplication of matrices
• Transpose of a matrix
• Symmetric and skew-symmetric matrices
• Invertible matrices

Matrices practice questions with answers

1. Q1Multiple choice·1 mark
   If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} x & 0 \\ 1 & 2 \end{pmatrix}$ are two matrices such that $AB = \begin{pmatrix} 4 & 4 \\ 10 & 8 \end{pmatrix}$, then the value of $x$ is:

   1. (a)$3$
   2. (b)$2$
   3. (c)$4$
   4. (d)$1$
   Show answer and step marks

   1. Correct option: (b) $2$+1

   Answer: (b)
2. Q2Multiple choice·1 mark
   If $A$ is a square matrix of order $3$ such that $|A| = 5$, then the value of $|3 A^T|$ is:

   1. (a)$405$
   2. (b)$15$
   3. (c)$45$
   4. (d)$135$
   Show answer and step marks

   1. Correct option: (d) $135$+1

   Answer: (d)
3. Q3Short answer·2 marks
   Let $A = \begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & -5 \end{pmatrix}$. Explain why $A$ is not a scalar matrix.
   Show answer and step marks

   1. A scalar matrix is a diagonal matrix in which all the diagonal elements are equal (so every non-diagonal element is zero and $a_{11} = a_{22} = a_{33}$).+1
   2. Here $a_{11} = a_{22} = 5$ but $a_{33} = -5 \neq 5$, so the diagonal elements are not all equal and $A$ is not a scalar matrix.+1

   Answer: A is not a scalar matrix because its diagonal elements are not all equal ($a_{33} = -5 \neq 5 = a_{11} = a_{22}$).
4. Q4Long answer·5 marks
   Let $A = \begin{pmatrix} x+2y & 3 \\ 5 & z-3 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & 2x-y \\ w & 7 \end{pmatrix}$ be two matrices such that $A = B$. (a) Form two equations from the first row of the matrices and solve them for $x$ and $y$. [2] (b) Find the values of $z$ and $w$. [1] (c) Let $C = \begin{pmatrix} 2 & -1 & 0 \\ 4 & 3 & 7 \end{pmatrix}$ and $D = \begin{pmatrix} 2 & 4 \\ -1 & 3 \\ 0 & 7 \end{pmatrix}$. Are the matrices $C$ and $D$ equal? Give a reason. [2]
   Show answer and step marks

   1. Equating the first-row entries of $A$ and $B$ gives $x + 2y = 4$ and $3 = 2x - y$.+1
   2. From $2x - y = 3$, $y = 2x - 3$; substituting in $x + 2y = 4$ gives $5x - 6 = 4$, so $x = 2$.+0.5
   3. Then $y = 2(2) - 3 = 1$, so $x = 2$ and $y = 1$.+0.5
   4. Equating the $(2,1)$ entries gives $w = 5$.+0.5
   5. Equating the $(2,2)$ entries gives $z - 3 = 7$, so $z = 10$.+0.5
   6. Matrix $C$ has order $2 \times 3$ and matrix $D$ has order $3 \times 2$.+1
   7. The orders differ, so $C$ and $D$ are not equal, since equal matrices must have the same order.+1

   Answer: (a) $x = 2$, $y = 1$. (b) $z = 10$, $w = 5$. (c) No, $C$ is of order $2 \times 3$ and $D$ is of order $3 \times 2$, so they cannot be equal.

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