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Given matrices A, B, and C, verify the associative property of matrix addition: (A+B)+C = A+(B+C).

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2 parts
a

Verified\text{Verified}

b

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Step-by-step solution

10 steps

Here are the steps to verify the given matrix properties:

Given matrices:

A=[3624],B=[−1718],C=[3419]A = \begin{bmatrix} 3 & 6 \\ 2 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & 7 \\ 1 & 8 \end{bmatrix}, \quad C = \begin{bmatrix} 3 & 4 \\ 1 & 9 \end{bmatrix}

Part a· 5 steps

Verify (A+B)+C=A+(B+C)(A+B)+C = A+(B+C)

  1. Calculate A+BA+B.

    A+B=[3624]+[−1718]=[3+(−1)6+72+14+8]=[213312]A+B = \begin{bmatrix} 3 & 6 \\ 2 & 4 \end{bmatrix} + \begin{bmatrix} -1 & 7 \\ 1 & 8 \end{bmatrix} = \begin{bmatrix} 3+(-1) & 6+7 \\ 2+1 & 4+8 \end{bmatrix} = \begin{bmatrix} 2 & 13 \\ 3 & 12 \end{bmatrix}

  2. Calculate (A+B)+C(A+B)+C.

    (A+B)+C=[213312]+[3419]=[2+313+43+112+9]=[517421](A+B)+C = \begin{bmatrix} 2 & 13 \\ 3 & 12 \end{bmatrix} + \begin{bmatrix} 3 & 4 \\ 1 & 9 \end{bmatrix} = \begin{bmatrix} 2+3 & 13+4 \\ 3+1 & 12+9 \end{bmatrix} = \begin{bmatrix} 5 & 17 \\ 4 & 21 \end{bmatrix}

  3. Calculate B+CB+C.

    B+C=[−1718]+[3419]=[−1+37+41+18+9]=[211217]B+C = \begin{bmatrix} -1 & 7 \\ 1 & 8 \end{bmatrix} + \begin{bmatrix} 3 & 4 \\ 1 & 9 \end{bmatrix} = \begin{bmatrix} -1+3 & 7+4 \\ 1+1 & 8+9 \end{bmatrix} = \begin{bmatrix} 2 & 11 \\ 2 & 17 \end{bmatrix}

  4. Calculate A+(B+C)A+(B+C).

    A+(B+C)=[3624]+[211217]=[3+26+112+24+17]=[517421]A+(B+C) = \begin{bmatrix} 3 & 6 \\ 2 & 4 \end{bmatrix} + \begin{bmatrix} 2 & 11 \\ 2 & 17 \end{bmatrix} = \begin{bmatrix} 3+2 & 6+11 \\ 2+2 & 4+17 \end{bmatrix} = \begin{bmatrix} 5 & 17 \\ 4 & 21 \end{bmatrix}

  5. Compare the results.

    Since [517421]=[517421]\begin{bmatrix} 5 & 17 \\ 4 & 21 \end{bmatrix} = \begin{bmatrix} 5 & 17 \\ 4 & 21 \end{bmatrix}, the statement (A+B)+C=A+(B+C)(A+B)+C = A+(B+C) is verified.
    Verified\boxed{\text{Verified}}

Part b

Verified\text{Verified}

Verify (A+B)−C=A+(B−C)(A+B)-C = A+(B-C)

  1. Calculate A+BA+B (from part a).

    A+B=[213312]A+B = \begin{bmatrix} 2 & 13 \\ 3 & 12 \end{bmatrix}

  2. Calculate (A+B)−C(A+B)-C.

    (A+B)−C=[213312]−[3419]=[2−313−43−112−9]=[−1923](A+B)-C = \begin{bmatrix} 2 & 13 \\ 3 & 12 \end{bmatrix} - \begin{bmatrix} 3 & 4 \\ 1 & 9 \end{bmatrix} = \begin{bmatrix} 2-3 & 13-4 \\ 3-1 & 12-9 \end{bmatrix} = \begin{bmatrix} -1 & 9 \\ 2 & 3 \end{bmatrix}

  3. Calculate B−CB-C.

    B−C=[−1718]−[3419]=[−1−37−41−18−9]=[−430−1]B-C = \begin{bmatrix} -1 & 7 \\ 1 & 8 \end{bmatrix} - \begin{bmatrix} 3 & 4 \\ 1 & 9 \end{bmatrix} = \begin{bmatrix} -1-3 & 7-4 \\ 1-1 & 8-9 \end{bmatrix} = \begin{bmatrix} -4 & 3 \\ 0 & -1 \end{bmatrix}

  4. Calculate A+(B−C)A+(B-C).

    A+(B−C)=[3624]+[−430−1]=[3+(−4)6+32+04+(−1)]=[−1923]A+(B-C) = \begin{bmatrix} 3 & 6 \\ 2 & 4 \end{bmatrix} + \begin{bmatrix} -4 & 3 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 3+(-4) & 6+3 \\ 2+0 & 4+(-1) \end{bmatrix} = \begin{bmatrix} -1 & 9 \\ 2 & 3 \end{bmatrix}

  5. Compare the results.

    Since [−1923]=[−1923]\begin{bmatrix} -1 & 9 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} -1 & 9 \\ 2 & 3 \end{bmatrix}, the statement (A+B)−C=A+(B−C)(A+B)-C = A+(B-C) is verified.
    Verified\boxed{\text{Verified}}

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