If A = [-121-32-3] and B = [21-32-13], prove that (A + BT)T = AT + B - Mathematics and Statistics

Sum

If A = `[(-1, 2, 1),(-3, 2, -3)]` and B = `[(2, 1),(-3, 2),(-1, 3)]`, prove that (A + B^T)^T = A^T + B

A = `[(-1, 2, 1),(-3, 2, -3)]` and B = `[(2, 1),(-3, 2),(-1, 3)]`

∴ A^T = `[(-1, -3),(2, 2),(1, -3)]` and B^T = `[(2, -3, -1),(1, 2, 3)]`

∴ A + B^T = `[(-1, 2, 1),(-3, 2, -3)] + [(2, -3, -1),(1, 2, 3)]`

= `[(-1 + 2, 2 - 3, 1 - 1),(-3 + 1, 2 + 2, -3 + 3)]`

= `[(1, -1, 0),(-2, 4, 0)]`

∴ (A + B^T)^T = `[(1, -2),(-1, 4),(0, 0)]` ...(i)

Now, A^T + B = `[(-1, -3),(2, 2),(1, -3)] + [(2, 1),(-3, 2),(-1, 3)]`

= `[(-1 + 2, -3 + 1),(2 - 3, 2 + 2),(1 - 1, -3 + 3)]`

= `[(1, -2),(-1, 4),(0, 0)]` ...(ii)

From (i) and (ii), we get

(A + B^T)^T = A^T + B

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Chapter 4 Determinants and Matrices
Exercise 4.7 | Q 10 | Page 98