Question

If for three matrices A = `[a_(ij)]_(m xx 4)`, B = `[b_(ij)]_(n xx 3)` and C = `[c_(ij)]_(p xx q)` products AB and AC both are defined and are square matrices of the same order, then the values of m, n, p and q are ______.

Options

• m = q = 3 and n = p = 4

• m = 2, q = 3 and n = p = 4

• m = q = 4 and n = p = 3

• m = 4, p = 2 and n = q = 3

Answer 1

If for three matrices A = `[a_(ij)]_(m xx 4)`, B = `[b_(ij)]_(n xx 3)` and C = `[c_(ij)]_(p xx q)` products AB and AC both are defined and are square matrices of the same order, then the values of m, n, p and q are m = q = 3 and n = p = 4.

Explanation:

Let’s check both AB and AC separately.

For AB:

Order of A is m × 4.

Order of B is n × 3.

AB = `[a_(ij)]_(m xx 4)  [b_(ij)]_(n xx 3)`

This is possible only if n is 4.

The order of AB is m × 3.

Since AB is a square matrix.

Both m and 3 should be equal.

∴ m = 3

For AC:

Order of A is m × 4.

Order of C is p × q.

AC = `[a_(ij)]_(m xx 4)  [c_(ij)]_(p xx q)`

This is possible only if p is 4.

The order of AC is m × q, i.e., 3 × q.

Since AC is a square matrix.

Both 3 and q should be equal.

∴ q = 3

Thus, m = q = 3 and n = p = 4.