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Given that A = [[1, 4], [2, 5]] and B = [[1, 1], [0, 2]]. Evaluate (I) AB (II) BA

Mathematics

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Given the matrices:

A=[1425]andB=[1102]A = \begin{bmatrix} 1 & 4 \\ 2 & 5 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix}

I) Evaluate ABAB

  1. Set up the multiplication for ABAB.

    AB=[1425][1102]AB = \begin{bmatrix} 1 & 4 \\ 2 & 5 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix}

  2. Perform the matrix multiplication.

    To find each element of ABAB, multiply the rows of AA by the columns of BB.

    AB=[(1)(1)+(4)(0)(1)(1)+(4)(2)(2)(1)+(5)(0)(2)(1)+(5)(2)]AB = \begin{bmatrix} (1)(1) + (4)(0) & (1)(1) + (4)(2) \\ (2)(1) + (5)(0) & (2)(1) + (5)(2) \end{bmatrix}

  3. Calculate the values for each element.

    AB=[1+01+82+02+10]AB = \begin{bmatrix} 1 + 0 & 1 + 8 \\ 2 + 0 & 2 + 10 \end{bmatrix}

  4. Simplify to get the final matrix for ABAB.

    AB=[19212]AB = \boxed{\begin{bmatrix} 1 & 9 \\ 2 & 12 \end{bmatrix}}

    II) Evaluate BABA

  5. Set up the multiplication for BABA.

    BA=[1102][1425]BA = \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix} \begin{bmatrix} 1 & 4 \\ 2 & 5 \end{bmatrix}

  6. Perform the matrix multiplication.

    To find each element of BABA, multiply the rows of BB by the columns of AA.

    BA=[(1)(1)+(1)(2)(1)(4)+(1)(5)(0)(1)+(2)(2)(0)(4)+(2)(5)]BA = \begin{bmatrix} (1)(1) + (1)(2) & (1)(4) + (1)(5) \\ (0)(1) + (2)(2) & (0)(4) + (2)(5) \end{bmatrix}

  7. Calculate the values for each element.

    BA=[1+24+50+40+10]BA = \begin{bmatrix} 1 + 2 & 4 + 5 \\ 0 + 4 & 0 + 10 \end{bmatrix}

  8. Simplify to get the final matrix for BABA.

    BA=[39410]BA = \boxed{\begin{bmatrix} 3 & 9 \\ 4 & 10 \end{bmatrix}}
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