Compute the indicated products. 1 2 3 2 3 4

To find: 1 2 3 2 3 4 Here order of the first matrix is 3 × 1 and the second matrix is 1 × 3 . As here the number of columns in the first matrix is same as number of rows in the second matrix, by using the multiplication principle of matrices we get, 1 2 3 2 3 4 = 1 × 2 1 × 3 1 × 4 2 × 2 2 × 3 2 × 4 3 × 2 3 × 3 3 × 4 = 2 3 4 4 6 8 6 9 12 Hence, 1 2 3 2 3 4 = 2 3 4 4 6 8 6 9 12

Express the matrix B = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 as the sum of symmetric and a skew-symmetric matrix.

Given that, B = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 To proof : symmetric and skew symmetric matrix. Let B = P + Q where P = 1 2 B + B ' and Q = 1 2 B - B ' Now, P = 1 2 B + B ' = 1 2 2 - 2 - 4 - 1 3 4 1 - 2 - 3 + 2 - 1 1 - 2 3 - 2 - 4 4 - 3 P = 1 2 B + B ' = 1 2 4 - 3 - 3 - 3 6 2 - 3 2 - 6 P = 1 2 B + B ' = 2 - 3 2 - 3 2 - 3 2 3 1 - 3 2 1 - 3 (symmetric) Q = 1 2 B - B ' = 1 2 2 - 2 - 4 - 1 3 4 1 - 2 - 3 - 2 - 1 1 - 2 3 - 2 - 4 4 - 3 Q = 1 2 B - B ' = 1 2 0 - 1 - 5 1 0 6 5 - 6 0 Q = 1 2 B - B ' = 0 - 1 2 - 5 2 1 2 0 3 5 2 - 3 0 (skew-symmetric) Hence, 2 - 2 - 4 - 1 3 4 1 - 2 - 3 = 2 - 3 2 - 3 2 - 3 2 3 1 - 3 2 1 - 3 + 0 - 1 2 - 5 2 1 2 0 3 5 2 - 3 0 .

Express the matrix B = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 as the sum of symmetric and a skew-symmetric matrix.

Given that, B = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 To proof : symmetric and skew symmetric matrix. Let B = P + Q where P = 1 2 B + B ' and Q = 1 2 B - B ' Now, P = 1 2 B + B ' = 1 2 2 - 2 - 4 - 1 3 4 1 - 2 - 3 + 2 - 1 1 - 2 3 - 2 - 4 4 - 3 P = 1 2 B + B ' = 1 2 4 - 3 - 3 - 3 6 2 - 3 2 - 6 P = 1 2 B + B ' = 2 - 3 2 - 3 2 - 3 2 3 1 - 3 2 1 - 3 (symmetric) Q = 1 2 B - B ' = 1 2 2 - 2 - 4 - 1 3 4 1 - 2 - 3 - 2 - 1 1 - 2 3 - 2 - 4 4 - 3 Q = 1 2 B - B ' = 1 2 0 - 1 - 5 1 0 6 5 - 6 0 Q = 1 2 B - B ' = 0 - 1 2 - 5 2 1 2 0 3 5 2 - 3 0 (skew-symmetric) Hence, 2 - 2 - 4 - 1 3 4 1 - 2 - 3 = 2 - 3 2 - 3 2 - 3 2 3 1 - 3 2 1 - 3 + 0 - 1 2 - 5 2 1 2 0 3 5 2 - 3 0 .

Compute the indicated products. 1 2 3 2 3 4

To find: 1 2 3 2 3 4 Here order of the first matrix is 3 × 1 and the second matrix is 1 × 3 . As here the number of columns in the first matrix is same as number of rows in the second matrix, by using the multiplication principle of matrices we get, 1 2 3 2 3 4 = 1 × 2 1 × 3 1 × 4 2 × 2 2 × 3 2 × 4 3 × 2 3 × 3 3 × 4 = 2 3 4 4 6 8 6 9 12 Hence, 1 2 3 2 3 4 = 2 3 4 4 6 8 6 9 12

Find a matrix A such that 2 A - 3 B + 5 C = 0 , where B = - 2 2 0 3 1 4 and C = 2 0 - 2 7 1 6

We have, 2 A - 3 B + 5 C = O ⇒ 2 A = 3 B - 5 C ⇒ 2 A = 3 - 2 2 0 3 1 4 - 5 2 0 - 2 7 1 6 2 A = - 6 6 0 9 3 12 - 10 0 - 10 35 5 30 2 A = - 16 6 10 - 26 - 2 - 18 ⇒ A = - 8 3 5 - 13 - 1 - 9