Chapter 4: Q10. (page 171)

Use $A = [\begin{matrix} 2 & - 1 \\ 3 & 5 \end{matrix}]$ , $B = [\begin{matrix} - 4 & 1 \\ 8 & 0 \end{matrix}]$ , and $C = [\begin{matrix} 3 & 2 \\ - 1 & 2 \end{matrix}]$ to determine whether $A (B C) = (A B) C$ is true for the given matrices.

Short Answer

Expert verified

Yes, the provided equation is true.

Step by step solution

Step 1- Determine the value of A(BC)

Substitute the provided matrices into . $A (B C)$

$A (B C) = [\begin{matrix} 2 & - 1 \\ 3 & 5 \end{matrix}] ([\begin{matrix} - 4 & 1 \\ 8 & 0 \end{matrix}] \cdot [\begin{matrix} 3 & 2 \\ - 1 & 2 \end{matrix}])$

First perform the multiplication between the matrices B and C after it multiply the resulting matrix with A.

$\begin{matrix} A (B C) = [\begin{matrix} 2 & - 1 \\ 3 & 5 \end{matrix}] \cdot [\begin{matrix} - 13 & - 6 \\ 24 & 16 \end{matrix}] \\ = [\begin{matrix} - 50 & - 28 \\ 81 & 62 \end{matrix}] \end{matrix}$

– Determine the value of (AB)C

Substitute the provided matrices into . $(A B) C$

$(A B) C = ([\begin{matrix} 2 & - 1 \\ 3 & 5 \end{matrix}] \cdot [\begin{matrix} - 4 & 1 \\ 8 & 0 \end{matrix}]) [\begin{matrix} 3 & 2 \\ - 1 & 2 \end{matrix}]$

First perform the multiplication between the matrices A and B after it multiply the resulting matrix with C.

$\begin{matrix} (A B) C = [\begin{matrix} - 16 & 2 \\ 28 & 3 \end{matrix}] \cdot [\begin{matrix} 3 & 2 \\ - 1 & 2 \end{matrix}] \\ = [\begin{matrix} - 50 & - 28 \\ 81 & 62 \end{matrix}] \end{matrix}$

Step 3- Make the conclusion

As both the sides results in same matrix which means that the provided equation is true.

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Use $A = [\begin{matrix} 2 & - 1 \\ 3 & 5 \end{matrix}]$ , $B = [\begin{matrix} - 4 & 1 \\ 8 & 0 \end{matrix}]$ , and $C = [\begin{matrix} 3 & 2 \\ - 1 & 2 \end{matrix}]$ to determine whether $A (B C) = (A B) C$ is true for the given matrices.

Short Answer

Step by step solution

Step 1- Determine the value of A(BC)

– Determine the value of (AB)C

Step 3- Make the conclusion

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UseA=[2−135],B=[−4180] , andC=[32−12] to determine whetherA(BC)=(AB)C is true for the given matrices.

Short Answer

Step by step solution

Step 1- Determine the value of A(BC)

– Determine the value of (AB)C

Step 3- Make the conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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Study anywhere. Anytime. Across all devices.

Use $A = [\begin{matrix} 2 & - 1 \\ 3 & 5 \end{matrix}]$ , $B = [\begin{matrix} - 4 & 1 \\ 8 & 0 \end{matrix}]$ , and $C = [\begin{matrix} 3 & 2 \\ - 1 & 2 \end{matrix}]$ to determine whether $A (B C) = (A B) C$ is true for the given matrices.