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Chapter 4: Problem 329

If $$ \mathrm{A}=\left|\begin{array}{cc} 1 & 0 \\ -1 & 7 \end{array}\right| $$ and \(A^{2}=8 \mathrm{~A}+\mathrm{kl}_{2}\), then \(\mathrm{k}=\ldots\) (a) 1 (b) \(-1\) (c) 7 (d) \(-7\)

Short Answer

Expert verified
The value of k is \(-7\) (Option d).

Step by step solution

01

STEP 1: Compute A^2

To find A^2, we need to multiply A by itself. Let's use the matrix multiplication rule: A^2 = A * A, where $$ \mathrm{A}=\left|\begin{array}{cc} 1 & 0 \\\ -1 & 7 \end{array}\right| $$ So, $$ A^2 = \left|\begin{array}{cc} 1 & 0 \\ -1 & 7 \end{array}\right| * \left|\begin{array}{cc} 1 & 0 \\ -1 & 7 \end{array}\right| $$
02

STEP 2: Use matrix multiplication rules

Now, let's perform matrix multiplication: $$ A^2 = \left|\begin{array}{cc} (1*1 + 0*(-1)) & (1*0 + 0*7) \\ (-1*1 + 7*(-1)) & (-1*0 + 7*7) \end{array}\right| $$
03

STEP 3: Compute the values

Now we can compute the values inside the matrix: $$ A^2 = \left|\begin{array}{cc} 1 & 0 \\ -8 & 49 \end{array}\right| $$
04

STEP 4: Write down the given equation

Next, let's write the given equation and substitute A^2 with the value we found: A^2 = 8A + kl_2 $$ \left|\begin{array}{cc} 1 & 0 \\ -8 & 49 \end{array}\right| = 8 \left|\begin{array}{cc} 1 & 0 \\ -1 & 7 \end{array}\right| + k \left|\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right| $$
05

STEP 5: Simplify the equation

Now, let's simplify the equation by multiplying the matrices by their respective constants: $$ \left|\begin{array}{cc} 1 & 0 \\ -8 & 49 \end{array}\right| = \left|\begin{array}{cc} 8 & 0 \\ -8 & 56 \end{array}\right| + \left|\begin{array}{cc} k & 0 \\ 0 & k \end{array}\right| $$
06

STEP 6: Find the value of k

By comparing the corresponding elements of the resulting matrices, we can easily find out the value of k: k = 1 - 8 => k = -7 Comparing with the options, we see that our solution matches option (d). Therefore, k = -7 (Option d).

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Most popular questions from this chapter

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