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Given that matrices A = [[2, 1, 8], [4, 2, 0], [6, 7, 8]] and B = [[1, 1, 1], [1, 2, 1], [0, 4, 3]], Evaluate (i) A^T B^T (ii) (AB)^T

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Given the matrices:

A=[218420678]andB=[111121043]A = \begin{bmatrix} 2 & 1 & 8 \\ 4 & 2 & 0 \\ 6 & 7 & 8 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 4 & 3 \end{bmatrix}

I) Evaluate ATBTA^T B^T

  1. Find the transposes of AA and BB.

    To find the transpose of a matrix, interchange its rows and columns.

    AT=[246127808]A^T = \begin{bmatrix} 2 & 4 & 6 \\ 1 & 2 & 7 \\ 8 & 0 & 8 \end{bmatrix} BT=[110124113]B^T = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 2 & 4 \\ 1 & 1 & 3 \end{bmatrix}

  2. Multiply ATA^T by BTB^T.

    ATBT=[246127808][110124113]A^T B^T = \begin{bmatrix} 2 & 4 & 6 \\ 1 & 2 & 7 \\ 8 & 0 & 8 \end{bmatrix} \begin{bmatrix} 1 & 1 & 0 \\ 1 & 2 & 4 \\ 1 & 1 & 3 \end{bmatrix} ATBT=[(2)(1)+(4)(1)+(6)(1)(2)(1)+(4)(2)+(6)(1)(2)(0)+(4)(4)+(6)(3)(1)(1)+(2)(1)+(7)(1)(1)(1)+(2)(2)+(7)(1)(1)(0)+(2)(4)+(7)(3)(8)(1)+(0)(1)+(8)(1)(8)(1)+(0)(2)+(8)(1)(8)(0)+(0)(4)+(8)(3)]A^T B^T = \begin{bmatrix} (2)(1)+(4)(1)+(6)(1) & (2)(1)+(4)(2)+(6)(1) & (2)(0)+(4)(4)+(6)(3) \\ (1)(1)+(2)(1)+(7)(1) & (1)(1)+(2)(2)+(7)(1) & (1)(0)+(2)(4)+(7)(3) \\ (8)(1)+(0)(1)+(8)(1) & (8)(1)+(0)(2)+(8)(1) & (8)(0)+(0)(4)+(8)(3) \end{bmatrix} ATBT=[2+4+62+8+60+16+181+2+71+4+70+8+218+0+88+0+80+0+24]A^T B^T = \begin{bmatrix} 2+4+6 & 2+8+6 & 0+16+18 \\ 1+2+7 & 1+4+7 & 0+8+21 \\ 8+0+8 & 8+0+8 & 0+0+24 \end{bmatrix} ATBT=[121634101229161624]A^T B^T = \boxed{\begin{bmatrix} 12 & 16 & 34 \\ 10 & 12 & 29 \\ 16 & 16 & 24 \end{bmatrix}}

    II) Evaluate (AB)T(AB)^T

  3. Calculate the product ABAB.

    AB=[218420678][111121043]AB = \begin{bmatrix} 2 & 1 & 8 \\ 4 & 2 & 0 \\ 6 & 7 & 8 \end{bmatrix} \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 4 & 3 \end{bmatrix} AB=[(2)(1)+(1)(1)+(8)(0)(2)(1)+(1)(2)+(8)(4)(2)(1)+(1)(1)+(8)(3)(4)(1)+(2)(1)+(0)(0)(4)(1)+(2)(2)+(0)(4)(4)(1)+(2)(1)+(0)(3)(6)(1)+(7)(1)+(8)(0)(6)(1)+(7)(2)+(8)(4)(6)(1)+(7)(1)+(8)(3)]AB = \begin{bmatrix} (2)(1)+(1)(1)+(8)(0) & (2)(1)+(1)(2)+(8)(4) & (2)(1)+(1)(1)+(8)(3) \\ (4)(1)+(2)(1)+(0)(0) & (4)(1)+(2)(2)+(0)(4) & (4)(1)+(2)(1)+(0)(3) \\ (6)(1)+(7)(1)+(8)(0) & (6)(1)+(7)(2)+(8)(4) & (6)(1)+(7)(1)+(8)(3) \end{bmatrix} AB=[2+1+02+2+322+1+244+2+04+4+04+2+06+7+06+14+326+7+24]AB = \begin{bmatrix} 2+1+0 & 2+2+32 & 2+1+24 \\ 4+2+0 & 4+4+0 & 4+2+0 \\ 6+7+0 & 6+14+32 & 6+7+24 \end{bmatrix} AB=[33627686135237]AB = \begin{bmatrix} 3 & 36 & 27 \\ 6 & 8 & 6 \\ 13 & 52 & 37 \end{bmatrix}

  4. Find the transpose of ABAB.

    (AB)T=[36133685227637](AB)^T = \begin{bmatrix} 3 & 6 & 13 \\ 36 & 8 & 52 \\ 27 & 6 & 37 \end{bmatrix} (AB)T=[36133685227637](AB)^T = \boxed{\begin{bmatrix} 3 & 6 & 13 \\ 36 & 8 & 52 \\ 27 & 6 & 37 \end{bmatrix}}
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