If A = ⎡ 1 2 3 ⎢ 3 1 2 ⎣ 2 3 1 ⎤ ⎥ ⎦ and B = ⎡ −5 7 1 ⎢ 1 −5 7 ⎣ 7 1 −5 ⎤ ⎥ ⎦ prove that A B = 18 I₃.

To prove that the product of matrices A and B equals 18 times the identity matrix I₃, we first need to calculate the product AB. Given:

Matrix A

A = ⎡ 1 2 3 ⎢ 3 1 2 ⎣ 2 3 1 ⎤ ⎥ ⎦

Matrix B

B = ⎡ −5 7 1 ⎢ 1 −5 7 ⎣ 7 1 −5 ⎤ ⎥ ⎦

Calculating the Product AB

We will multiply A and B using the matrix multiplication rules:

• The element in the i-th row and j-th column of the product is obtained by taking the dot product of the i-th row of A and the j-th column of B.

Step-by-Step Calculation

Let's compute each element of the resulting matrix AB:

• First row, first column: (1)(-5) + (2)(1) + (3)(7) = -5 + 2 + 21 = 18
• First row, second column: (1)(7) + (2)(-5) + (3)(1) = 7 - 10 + 3 = 0
• First row, third column: (1)(1) + (2)(7) + (3)(-5) = 1 + 14 - 15 = 0

• Second row, first column: (3)(-5) + (1)(1) + (2)(7) = -15 + 1 + 14 = 0
• Second row, second column: (3)(7) + (1)(-5) + (2)(1) = 21 - 5 + 2 = 18
• Second row, third column: (3)(1) + (1)(7) + (2)(-5) = 3 + 7 - 10 = 0

• Third row, first column: (2)(-5) + (3)(1) + (1)(7) = -10 + 3 + 7 = 0
• Third row, second column: (2)(7) + (3)(-5) + (1)(1) = 14 - 15 + 1 = 0
• Third row, third column: (2)(1) + (3)(7) + (1)(-5) = 2 + 21 - 5 = 18

Resulting Matrix AB

The resulting matrix from the multiplication is:

AB = ⎡ 18 0 0 ⎢ 0 18 0 ⎣ 0 0 18 ⎤ ⎥ ⎦

Final Verification

This can be expressed as:

AB = 18 * I₃, where I₃ is the 3x3 identity matrix:

I₃ = ⎡ 1 0 0 ⎢ 0 1 0 ⎣ 0 0 1 ⎤ ⎥ ⎦

Thus, we have successfully proven that AB = 18 I₃.