Q1 / 120 · NDA 2024 (Phase 2) · Mathematics

Let X be a matrix of order $3 \times 3, \mathrm{Y}$ be a matrix of order $2 \times 3$ and Z be a matrix of order $3 \times 2$. Which of the following statements are correct? A. $(\mathrm{ZY}) \mathrm{X}$ is defined and is a square matrix of order 3 . B. $\mathrm{Y}(\mathrm{XZ})$ is defined and is a square matrix of order 2. C. $\mathrm{X}(\mathrm{YZ})$ is not defined. Select the answer using the code given below.

  • A
    $A$ and $B$ only
  • B
    B and C only
  • C
    A and C only
  • Correct answer
    D
    $\mathrm{A}, \mathrm{B}$ and C

Solution

Since we know that matrix multiplication is only possible when column of first matrix is equal to row of the second matrix. $A_{m \times n} \cdot B_{n \times p}=(A B)_{m \times p}$ Thus $\left[\mathrm{Z}_{3 \times 2} \cdot \mathrm{Y}_{2 \times 3}\right] \mathrm{X}_{3 \times 3}=(\mathrm{zyx})_{3 \times 3}$ and $\mathrm{Y}_{2 \times 3} \cdot\left[\mathrm{X}_{3 \times 3} \cdot \mathrm{Z}_{3 \times 2}\right]=(\mathrm{yxz})_{2 \times 2}$ and $\mathrm{x}_{3 \times 3}\left[\mathrm{Y}_{2 \times 3} \cdot \mathrm{Z}_{3 \times 2}\right]=\mathrm{X}_{3 \times 3} \cdot(\mathrm{YZ})_{2 \times 2}$ $\because$ No. of columns in $\mathrm{X}=$ no. of rows in $(\mathrm{yz})$ Hence $x(y z)$ is not defined.

Hence, the correct answer is option: 'D'.

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