Math Problem Statement

​Given​ M, find Upper M Superscript negative 1M−1 and show that Upper M Superscript negative 1M−1Mequals=I. Upper MMequals=left bracket Start 2 By 2 Matrix 1st Row 1st Column negative 1 2nd Column 0 2nd Row 1st Column 4 2nd Column 1 EndMatrix right bracket −1 0 4 1 Question content area bottom Part 1 Find Upper M Superscript negative 1M−1. Upper M Superscript negative 1M−1equals=negative 1−1 Part 2 Find the value in the first row and first column of the product Upper M Superscript negative 1M−1M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. ​(4 times •negative 1−1​)plus+​(11 times •4​)equals=enter your response here ​(Simplify your​ answer.) B. ​(4 times •00​)plus+​(11 times •1​)equals=enter your response here ​(Simplify your​ answer.) C. ​(negative 1−1 times •negative 1−1​)plus+​(00 times •4​)equals=enter your response here ​(Simplify your​ answer.) D. ​(negative 1−1 times •00​)plus+​(00 times •1​)equals=enter your response here ​(Simplify your​ answer.) Part 3 Find the value in the first row and second column of the product Upper M Superscript negative 1M−1M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. ​(negative 1−1 times •negative 1−1​)plus+​(00 times •4​)equals=enter your response here ​(Simplify your​ answer.) B. ​(4 times •negative 1−1​)plus+​(11 times •4​)equals=enter your response here ​(Simplify your​ answer.) C. ​(negative 1−1 times •00​)plus+​(00 times •1​)equals=enter your response here ​(Simplify your​ answer.) D. ​(4 times •00​)plus+​(11 times •1​)equals=enter your response here ​(Simplify your​ answer.) Part 4 Find the value in the second row and first column of the product Upper M Superscript negative 1M−1M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. ​(negative 1−1 times •00​)plus+​(00 times •1​)equals=enter your response here ​(Simplify your​ answer.) B. ​(4 times •00​)plus+​(1 times •1​)equals=enter your response here ​(Simplify your​ answer.) C. ​(negative 1−1 times •negative 1−1​)plus+​(00 times •4​)equals=enter your response here ​(Simplify your​ answer.) D. ​(4 times •negative 1−1​)plus+​(1 times •4​)equals=enter your response here ​(Simplify your​ answer.) Part 5 Find the value in the second row and second column of the product Upper M Superscript negative 1M−1M using matrix multiplication. Select the correct expression below and fill in the answer box to complete your selection. A. ​(4 times •00​)plus+​(1 times •1​)equals=enter your response here ​(Simplify your​ answer.) B. ​(negative 1−1 times •negative 1−1​)plus+​(00 times •4​)equals=enter your response here ​(Simplify your​ answer.) C. ​(4 times •negative 1−1​)plus+​(1 times •4​)equals=enter your response here ​(Simplify your​ answer.) D. ​(negative 1−1 times •00​)plus+​(00 times •1​)equals=enter your response here ​(Simplify your​ answer.) Part 6 When these values are shown in a​ matrix, the result is enter your response here​, showing that Upper M Superscript negative 1M−1Mequals=I.

Solution

Part 1: Finding M−1M^{-1}

Given matrix M=[−1041]M = \begin{bmatrix} -1 & 0 \\ 4 & 1 \end{bmatrix}, we will first find its inverse, M−1M^{-1}.

For a 2x2 matrix M=[abcd]M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, the inverse M−1M^{-1} is given by:

M−1=1ad−bc[d−b−ca]M^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

For M=[−1041]M = \begin{bmatrix} -1 & 0 \\ 4 & 1 \end{bmatrix}, we have:

  • a=−1a = -1, b=0b = 0, c=4c = 4, and d=1d = 1.

The determinant ad−bc=(−1)(1)−(0)(4)=−1ad - bc = (-1)(1) - (0)(4) = -1.

Thus, the inverse is:

M−1=1−1[10−4−1]=[−1041]M^{-1} = \frac{1}{-1} \begin{bmatrix} 1 & 0 \\ -4 & -1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 4 & 1 \end{bmatrix}

Part 2: First Row, First Column of M−1MM^{-1}M

We now multiply M−1M^{-1} by MM, starting with the first row and first column. This involves calculating:

First row, first column=(−1×−1)+(0×4)=1\text{First row, first column} = (-1 \times -1) + (0 \times 4) = 1

So, the expression is:

(−1×−1)+(0×4)=1(−1 × −1) + (0 × 4) = 1

This matches option C.

Part 3: First Row, Second Column of M−1MM^{-1}M

For the first row and second column, we calculate:

First row, second column=(−1×0)+(0×1)=0\text{First row, second column} = (-1 \times 0) + (0 \times 1) = 0

So, the expression is:

(−1×0)+(0×1)=0(−1 × 0) + (0 × 1) = 0

This matches option D.

Part 4: Second Row, First Column of M−1MM^{-1}M

For the second row and first column, we calculate:

Second row, first column=(4×−1)+(1×4)=−4+4=0\text{Second row, first column} = (4 \times -1) + (1 \times 4) = -4 + 4 = 0

So, the expression is:

(4×−1)+(1×4)=0(4 × −1) + (1 × 4) = 0

This matches option D.

Part 5: Second Row, Second Column of M−1MM^{-1}M

For the second row and second column, we calculate:

Second row, second column=(4×0)+(1×1)=0+1=1\text{Second row, second column} = (4 \times 0) + (1 \times 1) = 0 + 1 = 1

So, the expression is:

(4×0)+(1×1)=1(4 × 0) + (1 × 1) = 1

This matches option A.

Part 6: Matrix Result

The resulting matrix after multiplying M−1MM^{-1}M is:

[1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Thus, we have shown that M−1M=IM^{-1}M = I, where II is the identity matrix.

Do you have any questions or would you like more details? Here are some related questions to explore:

  1. How do you find the determinant of a 2x2 matrix?
  2. What is the significance of the identity matrix in linear algebra?
  3. Can you apply the inverse matrix method to solve systems of equations?
  4. How do you compute the inverse of a 3x3 matrix?
  5. Why is matrix multiplication not commutative in general?

Tip: Always check the determinant before computing a matrix inverse, as a zero determinant means the matrix is not invertible!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inversion
Matrix Multiplication

Formulas

Inverse of a 2x2 matrix: M^{-1} = (1/(ad-bc)) * [[d, -b], [-c, a]]
Matrix multiplication: [A] * [B] = [C]

Theorems

Identity Matrix Theorem
Inverse Matrix Properties

Suitable Grade Level

Grades 11-12 (Pre-Calculus, Early College Level)

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