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Chapter 9: Problem 58

Explain how to find the multiplicative inverse for a \(2 \times 2\) invertible matrix.

Short Answer

Expert verified
The multiplicative inverse of a \(2 \times 2\) matrix \(A = \begin{bmatrix}a & b\ c & d\end{bmatrix}\) is \(A^{-1} = \frac{1}{ad-bc} \times \begin{bmatrix}d & -b\ -c & a\end{bmatrix}\).

Step by step solution

01

Calculate the Determinant

Assume the matrix as \(A = \begin{bmatrix}a & b\ c & d\end{bmatrix}\). The determinant (det \(A\)) is calculated as: \[det(A) = ad - bc\]. Note that the determinant should not be equal to 0. If the determinant is 0, the matrix does not have a multiplicative inverse.
02

Finding the Adjugate

To find the adjugate, swap the a and d elements while changing the sign of b and c. This results in the adjugate matrix \(adj(A)\), \[\begin{bmatrix}d & -b\ -c & a\end{bmatrix}\].
03

Calculate the Multiplicative Inverse

The multiplicative inverse of a matrix \( A^{-1}\) is obtained by dividing the adjugate matrix by the determinant as follows: \(A^{-1} = \frac{1}{det(A)} \times adj(A) = \frac{1}{ad-bc} \times \begin{bmatrix}d & -b\ -c & a\end{bmatrix}\). Thus, you get the multiplicative inverse of matrix A.\n

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Matrix row operations remind me of what I did when solving a linear system by the addition method, although I no longer write the variables.

Let $$ A=\left[\begin{array}{rr} {-3} & {-7} \\ {2} & {-9} \\ {5} & {0} \end{array}\right] \text { and } B=\left[\begin{array}{rr} {-5} & {-1} \\ {0} & {0} \\ {3} & {-4} \end{array}\right] $$ Solve each matrix equation for X. $$ X-B=A $$

Consider a square matrix such that each element that is not on the diagonal from upper left to lower right is zero. Experiment with such matrices (call each matrix \(A\) ) by finding \(A A .\) Then write a sentence or two describing a method for multiplying this kind of matrix by itself.

Let $$ A=\left[\begin{array}{rr} {-3} & {-7} \\ {2} & {-9} \\ {5} & {0} \end{array}\right] \text { and } B=\left[\begin{array}{rr} {-5} & {-1} \\ {0} & {0} \\ {3} & {-4} \end{array}\right] $$ Solve each matrix equation for X. $$ 4 B+3 A=-2 X $$

What are equal matrices?
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