Questions: Given M, find M^-1 and show that M^-1 M=I. M=[ -1 0 8 1 ] Find M^-1. M^-1=[ ] Find the value in the first row and first column of the product M^-1 M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. (8 * 0)+(1 * 1)= [ ] (Simplify your answer.) B. (-1 * 0)+(0 * 1)= [ ] (Simplify your answer.) C. (8 * -1)+(1 * 8)= [ ] (Simplify your answer.) D. (-1 * -1)+(0 * 8)= [ ] (Simplify your answer.) Find the value in the first row and second column of the product M^-1 M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. (-1 * -1)+(0 * 8)= [ ] (Simplify your answer.) B. (8 * -1)+(1 * 8)= [ ] (Simplify your answer.) C. (8 * 0)+(1 * 1)= [ ] (Simplify your answer.) D. (-1 * 0)+(0 * 1)= [ ] (Simplify your answer.) Find the value in the second row and first column of the product M^-1 M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. (8 * -1)+(1 * 8)= [ ] (Simplify your answer.) B. (-1 * -1)+(0 * 8)= [ ] (Simplify your answer.) C. (8 * 0)+(1 * 1)= [ ] (Simplify your answer.) D. (-1 * 0)+(0 * 1)= [ ] (Simplify your answer.) Find the value in the second row and second column of the product M^-1 M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. (-1 * -1)+(0 * 8)= [ ] (Simplify your answer.) B. (8,-1)+(1,8)= [ ] (Simplify your answer.) C. (-1,0)+(0,1)= [ ] (Simplify your answer.) D. (8 * 0)+(1 * 1)= [ ] (Simplify your answer.) [ ]

Given M, find M^-1 and show that M^-1 M=I. M=[ -1 0 8 1 ] Find M^-1. M^-1=[ ] Find the value in the first row and first column of the product M^-1 M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. (8 * 0)+(1 * 1)= [ ] (Simplify your answer.) B. (-1 * 0)+(0 * 1)= [ ] (Simplify your answer.) C. (8 * -1)+(1 * 8)= [ ] (Simplify your answer.) D. (-1 * -1)+(0 * 8)= [ ] (Simplify your answer.) Find the value in the first row and second column of the product M^-1 M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. (-1 * -1)+(0 * 8)= [ ] (Simplify your answer.) B. (8 * -1)+(1 * 8)= [ ] (Simplify your answer.) C. (8 * 0)+(1 * 1)= [ ] (Simplify your answer.) D. (-1 * 0)+(0 * 1)= [ ] (Simplify your answer.) Find the value in the second row and first column of the product M^-1 M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. (8 * -1)+(1 * 8)= [ ] (Simplify your answer.) B. (-1 * -1)+(0 * 8)= [ ] (Simplify your answer.) C. (8 * 0)+(1 * 1)= [ ] (Simplify your answer.) D. (-1 * 0)+(0 * 1)= [ ] (Simplify your answer.) Find the value in the second row and second column of the product M^-1 M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. (-1 * -1)+(0 * 8)= [ ] (Simplify your answer.) B. (8,-1)+(1,8)= [ ] (Simplify your answer.) C. (-1,0)+(0,1)= [ ] (Simplify your answer.) D. (8 * 0)+(1 * 1)= [ ] (Simplify your answer.) [ ]
Transcript text: Given M , find $\mathrm{M}^{-1}$ and show that $\mathrm{M}^{-1} \mathrm{M}=\mathrm{I}$. \[ M=\left[\begin{array}{rr} -1 & 0 \\ 8 & 1 \end{array}\right] \] Find $\mathrm{M}^{-1}$. \[ \mathrm{M}^{-1}=\square \] Find the value in the first row and first column of the product $\mathrm{M}^{-1} \mathrm{M}$ using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. $(8 \cdot 0)+(1 \cdot 1)=\square$ (Simplify your answer.) $\square$ B. $(-1 \cdot 0)+(0 \cdot 1)=$ $\square$ (Simplify your answer.) C. $(8 \cdot-1)+(1 \cdot 8)=$ $\square$ (Simplify your answer.) D. $(-1 \cdot-1)+(0 \cdot 8)=$ $\square$ (Simplify your answer.) Find the value in the first row and second column of the product $\mathrm{M}^{-1} \mathrm{M}$ using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. $(-1 \cdot-1)+(0 \cdot 8)=$ $\square$ (Simplify your answer.) B. $(8 \cdot-1)+(1 \cdot 8)=\square$ (Simplify your answer.) C. $(8 \cdot 0)+(1 \cdot 1)=\square$ (Simplify your answer.) D. $(-1 \cdot 0)+(0 \cdot 1)=\square$ (Simplify your answer.) Find the value in the second row and first column of the product $M^{-1} M$ using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. $(8 \cdot-1)+(1 \cdot 8)=\square$ (Simplify your answer.) B. $(-1 \cdot-1)+(0 \cdot 8)=\square$ (Simplify your answer.) C. $(8 \cdot 0)+(1 \cdot 1)=$ $\square$ (Simplify your answer.) D. $(-1 \cdot 0)+(0 \cdot 1)=\square$ (Simplify your answer.) Find the value in the second row and second column of the product $\mathrm{M}^{-1} \mathrm{M}$ using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. $(-1 \cdot-1)+(0 \cdot 8)=$ $\square$ (Simplify your answer.) B. $(8,-1)+(1,8)=$ $\square$ (Simplify your answer.) C. $(-1,0)+(0,1)=$ $\square$ (Simplify your answer.) D. $(8 \cdot 0)+(1 \cdot 1)=\square$ (Simplify your answer.) $\square$
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Solution

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Solution Steps

Step 1: Define the Matrix \( M \)

Let \( M = \begin{bmatrix} -1 & 0 \\ 8 & 1 \end{bmatrix} \).

Step 2: Calculate the Inverse \( M^{-1} \)

The inverse of matrix \( M \) is calculated as follows: \[ M^{-1} = \begin{bmatrix} -1 & 0 \\ 8 & 1 \end{bmatrix} \]

Step 3: Verify the Product \( M^{-1} M \)

Now, we compute the product \( M^{-1} M \): \[ M^{-1} M = \begin{bmatrix} -1 & 0 \\ 8 & 1 \end{bmatrix} \begin{bmatrix} -1 & 0 \\ 8 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]

Step 4: Extract Specific Elements of the Product Matrix

From the product matrix \( M^{-1} M \):

  • The value in the first row and first column is \( 1 \).
  • The value in the first row and second column is \( 0 \).
  • The value in the second row and first column is \( 0 \).
  • The value in the second row and second column is \( 1 \).
Step 5: Conclusion

The product \( M^{-1} M \) results in the identity matrix \( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \), confirming that \( M^{-1} M = I \).

Final Answer

\(\mathrm{M}^{-1} = \begin{bmatrix} -1 & 0 \\ 8 & 1 \end{bmatrix}\)

First row, first column: \(\boxed{1}\)

First row, second column: \(\boxed{0}\)

Second row, first column: \(\boxed{0}\)

Second row, second column: \(\boxed{1}\)

Thus, \(\mathrm{M}^{-1} \mathrm{M} = \mathrm{I}\).

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