Chapter 4: Q27. (page 172)

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

Short Answer

Expert verified

The provided equation is true.

Step by step solution

Step 1- Determine the value of AC+BC

Substitute the provided matrices into $A C + B C$

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

First perform the multiplication after it perform addition.

$\begin{matrix} A C + B C = [\begin{matrix} 1 & 9 \\ 26 & - 8 \end{matrix}] + [\begin{matrix} - 21 & - 13 \\ 26 & - 8 \end{matrix}] \\ = [\begin{matrix} - 20 & - 4 \\ 52 & - 16 \end{matrix}] \end{matrix}$

– Determine the value of (A+B)C

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

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

First perform the addition after it multiplication.

$\begin{matrix} (A + B) C = [\begin{matrix} - 4 & 0 \\ 8 & 6 \end{matrix}] \cdot [\begin{matrix} 5 & 1 \\ 2 & - 4 \end{matrix}] \\ = [\begin{matrix} - 20 & - 4 \\ 52 & - 16 \end{matrix}] \end{matrix}$

Step 3- Make the conclusion

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

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

Short Answer

Step by step solution

Step 1- Determine the value of AC+BC

– Determine the value of (A+B)C

Step 3- Make the conclusion

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UseA=[1−243],B=[−5243], and C=[512−4] and scalar c=3to determine whether the equation is true for the given matrices.AC+BC=(A+B)C

Short Answer

Step by step solution

Step 1- Determine the value of AC+BC

– Determine the value of (A+B)C

Step 3- Make the conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

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