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Given matrices A and B, evaluate AB, BA, and A(AB).

Mathematics

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3 parts
i

(462412−2)\begin{pmatrix} 46 & 24 \\ 12 & -2 \end{pmatrix}

ii

(4301840)\begin{pmatrix} 4 & 30 \\ 18 & 40 \end{pmatrix}

iii

(338102−44−56)\begin{pmatrix} 338 & 102 \\ -44 & -56 \end{pmatrix}

Step-by-step solution

7 steps

Given the matrices:

A=(59−24)A = \begin{pmatrix} 5 & 9 \\ -2 & 4 \end{pmatrix} B=(2341)B = \begin{pmatrix} 2 & 3 \\ 4 & 1 \end{pmatrix}

Part i· 2 steps

Evaluate AB

  1. Multiply matrix A by matrix B.

    AB=(59−24)(2341)AB = \begin{pmatrix} 5 & 9 \\ -2 & 4 \end{pmatrix} \begin{pmatrix} 2 & 3 \\ 4 & 1 \end{pmatrix}

  2. Perform the matrix multiplication.

    AB=((5)(2)+(9)(4)(5)(3)+(9)(1)(−2)(2)+(4)(4)(−2)(3)+(4)(1))AB = \begin{pmatrix} (5)(2) + (9)(4) & (5)(3) + (9)(1) \\ (-2)(2) + (4)(4) & (-2)(3) + (4)(1) \end{pmatrix} AB=(10+3615+9−4+16−6+4)AB = \begin{pmatrix} 10 + 36 & 15 + 9 \\ -4 + 16 & -6 + 4 \end{pmatrix} AB=(462412−2)AB = \begin{pmatrix} 46 & 24 \\ 12 & -2 \end{pmatrix}

    The result for AB is:

    (462412−2)\boxed{\begin{pmatrix} 46 & 24 \\ 12 & -2 \end{pmatrix}}

Part ii

(4301840)\begin{pmatrix} 4 & 30 \\ 18 & 40 \end{pmatrix}

Evaluate BA

  1. Multiply matrix B by matrix A.

    BA=(2341)(59−24)BA = \begin{pmatrix} 2 & 3 \\ 4 & 1 \end{pmatrix} \begin{pmatrix} 5 & 9 \\ -2 & 4 \end{pmatrix}

  2. Perform the matrix multiplication.

    BA=((2)(5)+(3)(−2)(2)(9)+(3)(4)(4)(5)+(1)(−2)(4)(9)+(1)(4))BA = \begin{pmatrix} (2)(5) + (3)(-2) & (2)(9) + (3)(4) \\ (4)(5) + (1)(-2) & (4)(9) + (1)(4) \end{pmatrix} BA=(10−618+1220−236+4)BA = \begin{pmatrix} 10 - 6 & 18 + 12 \\ 20 - 2 & 36 + 4 \end{pmatrix} BA=(4301840)BA = \begin{pmatrix} 4 & 30 \\ 18 & 40 \end{pmatrix}

    The result for BA is:

    (4301840)\boxed{\begin{pmatrix} 4 & 30 \\ 18 & 40 \end{pmatrix}}

Part iii

(338102−44−56)\begin{pmatrix} 338 & 102 \\ -44 & -56 \end{pmatrix}

Evaluate A(AB)

  1. Use the result of AB from part (i).

    AB=(462412−2)AB = \begin{pmatrix} 46 & 24 \\ 12 & -2 \end{pmatrix}

  2. Multiply matrix A by the result of AB.

    A(AB)=(59−24)(462412−2)A(AB) = \begin{pmatrix} 5 & 9 \\ -2 & 4 \end{pmatrix} \begin{pmatrix} 46 & 24 \\ 12 & -2 \end{pmatrix}

  3. Perform the matrix multiplication.

    A(AB)=((5)(46)+(9)(12)(5)(24)+(9)(−2)(−2)(46)+(4)(12)(−2)(24)+(4)(−2))A(AB) = \begin{pmatrix} (5)(46) + (9)(12) & (5)(24) + (9)(-2) \\ (-2)(46) + (4)(12) & (-2)(24) + (4)(-2) \end{pmatrix} A(AB)=(230+108120−18−92+48−48−8)A(AB) = \begin{pmatrix} 230 + 108 & 120 - 18 \\ -92 + 48 & -48 - 8 \end{pmatrix} A(AB)=(338102−44−56)A(AB) = \begin{pmatrix} 338 & 102 \\ -44 & -56 \end{pmatrix}

    The result for A(AB) is:

    (338102−44−56)\boxed{\begin{pmatrix} 338 & 102 \\ -44 & -56 \end{pmatrix}}
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