Let X be a matrix of order 3 x 3, Y be a matrix of order 2 x 3 and Z be a matrix of order 3 × 2. Which of the following statements are correct?

I. (ZY)X is defined and is a square matrix of order 3.

II. Y(XZ) is defined and is a square matrix of order 2.

III. X(YZ) is not defined.

Select the answer using the code given below.

Concept:

Matrix Multiplication:

• Matrix multiplication is defined only when the number of columns in the first matrix is equal to the number of rows in the second matrix.
• For two matrices \( A_{m \times n} \) and \( B_{n \times p} \), their product \( AB \) will result in a matrix \( C_{m \times p} \).
• In general, the product of \( X \) and \( Y \) is defined if and only if the number of columns in \( X \) equals the number of rows in \( Y \).

Calculation:

We have the following matrix operations:

\( A_{m \times n} B_{n \times p} = (AB)_{m \times p} \)

Now, consider the matrix multiplications:

\([Z_{3 \times 2} . Y_{2\times3}].X_{3 \times 3}] = [ZYX]_{3 \times 3}\)

\( Y_{2 \times 3} [X_{3 \times 3} Z_{3 \times 2}] = [YXZ]_{2 \times 2} \)

Finally, we observe that:

\( X_{3 \times 3} [Y_{2 \times 3} Z_{3 \times 2}] = X_{3 \times 3} [YZ]_{2 \times 2} \)

Conclusion:

\(\text{No. of columns in X} \neq \text{No. of rows in (YZ)}\)

Hence, X(YZ) is not defined.

∴ the Correct answer is Option 4