If P = [(x,0,0),(0,y,0),(0,0,z)], Q = [(a,0,0),(0,b,0),(0,0,c)], prove that PQ

Question

If P = \(\begin{bmatrix} x & 0 & 0 \\[0.3em] 0 & y & 0 \\[0.3em] 0 & 0 & z \end{bmatrix}\), Q = \(\begin{bmatrix} a & 0 & 0 \\[0.3em] 0 & b & 0 \\[0.3em] 0 & 0 & c \end{bmatrix}\), prove that PQ = \(\begin{bmatrix} xa & 0 & 0 \\[0.3em] 0 & yb & 0 \\[0.3em] 0 & 0 & zc \end{bmatrix}\) = QP

Answer 1

Given: We have matrices P and Q, such that

Since, in order to multiply two matrices, A and B, the number of columns in A must equal the number of rows in B. Thus, if A is an m x n matrix and B is an r x s matrix, n = r.

Order of P = 3 × 3

And order of Q = 3 × 3

Number of columns of matrix P = Number of rows of matrix Q = 3

So, P and Q can be multiplied.

So, multiply 1^st row of matrix P by matching members of 1^st column of matrix Q, then sum them up.

(x, 0, 0)(a, 0, 0) = (x × a) + (0 × 0) + (0 × 0)

⇒ (x, 0, 0)(a, 0, 0) = xa

Multiply 1^st row of matrix P by matching members of 2^nd column of matrix Q, then sum them up.

(x, 0, 0)(0, b, 0) = (x × 0) + (0 × b) + (0 × 0)

⇒ (x, 0, 0)(0, b, 0) = 0

Similarly, repeat the steps to find other elements.

So,

Multiply 1^st row of matrix Q by matching members of 1^st column of matrix P, then sum them up.

(a, 0, 0)(x, 0, 0) = (a × x) + (0 × 0) + (0 × 0)

⇒ (a, 0, 0)(x, 0, 0) = xa + 0 + 0

⇒ (a, 0, 0)(x, 0, 0) = xa