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Chapter 9: Problem 30

Find the inverse of each matrix \(A\) if possible. Check that \(A A^{-1}=I\) and \(A^{-1} A=I .\) See the procedure for finding \(A^{-1}\) $$\left[\begin{array}{rr}4 & 3 \\ -3 & -2\end{array}\right]$$

Short Answer

Expert verified
The inverse of the matrix \(A\) is \(\left[\begin{array}{rr}-2 & -3 \ 3 & 4\end{array}\right]\).

Step by step solution

01

Calculate the Determinant

Find the determinant of matrix \(A\). \(A = \left[\begin{array}{rr}4 & 3 \ -3 & -2\end{array}\right]\). The determinant is calculated as: \( \text{det}(A) = (4)(-2) - (3)(-3) = -8 + 9 = 1 \).
02

Check if the Determinant is non-zero

Since the determinant is \(1\), which is non-zero, the inverse of the matrix exists.
03

Use the Formula for the Inverse of a 2x2 Matrix

The formula for the inverse of a 2x2 matrix \( \left[\begin{array}{rr}a & b \ c & d\end{array}\right] \) is given by: \( A^{-1} = \frac{1}{\text{det}(A)} \left[ \begin{array}{rr}d & -b \ -c & a\end{array}\right] \). Substitute the values: \( A^{-1} = \left[ \begin{array}{rr}-2 & -3 \ 3 & 4\end{array}\right] \).
04

Simplify the Inverse Matrix

Since the determinant is 1, the inverse matrix does not require further division: \( A^{-1} = \left[\begin{array}{rr}-2 & -3 \ 3 & 4\end{array}\right] \).
05

Verify that \(A A^{-1} = I\)

Calculate \(A A^{-1}\): \( \left[\begin{array}{rr}4 & 3 \ -3 & -2\end{array}\right] \left[\begin{array}{rr}-2 & -3 \ 3 & 4\end{array}\right] = \left[\begin{array}{rr}(4)(-2) + (3)(3) & (4)(-3) + (3)(4) \ (-3)(-2) + (-2)(3) & (-3)(-3) + (-2)(4)\end{array}\right] = \left[\begin{array}{rr}1 & 0 \ 0 & 1\end{array}\right] \). Hence, \(A A^{-1}=I\).
06

Verify that \(A^{-1} A = I\)

Calculate \(A^{-1} A\): \( \left[\begin{array}{rr}-2 & -3 \ 3 & 4\end{array}\right] \left[\begin{array}{rr}4 & 3 \ -3 & -2\end{array}\right] = \left[\begin{array}{rr}(-2)(4) + (-3)(-3) & (-2)(3) + (-3)(-2) \ (3)(4) + (4)(-3) & (3)(3) + (4)(-2)\end{array}\right] = \left[\begin{array}{rr}1 & 0 \ 0 & 1\end{array}\right] \). Hence, \(A^{-1} A = I\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Determinant Calculation
The determinant of a matrix provides vital information about the matrix, including whether it has an inverse. For a 2x2 matrix \(A = \left[\begin{array}{rr}a & b \ c & d\end{array}\right]\), the determinant is calculated using the formula:
\[ \text{det}(A) = (a \cdot d) - (b \cdot c) \]
In the given exercise, the matrix \(A\) is \(\left[\begin{array}{rr}4 & 3 \ -3 & -2\end{array}\right]\). Plugging the values into the formula, we get:

\[ \text{det}(A) = (4 \cdot -2) - (3 \cdot -3) = -8 + 9 = 1 \]
Since the determinant is 1, which is non-zero, the matrix has an inverse.
2x2 Matrix
A 2x2 matrix is a simple type of matrix with two rows and two columns, represented as:
\( \left[\begin{array}{rr}a & b \ c & d\end{array}\right] \). Each element in the matrix is situated within a specific row and column. In the example matrix \(A\), we have:
  • 4 and 3 in the first row
  • -3 and -2 in the second row
Understanding the layout helps in performing operations like finding determinants and calculating inverses.
Identity Matrix
The identity matrix is a special kind of matrix which doesn鈥檛 change any vector when we multiply the vector by it. It is analogous to the number 1 in multiplication. For a 2x2 matrix, the identity matrix is:
\( I = \left[\begin{array}{rr}1 & 0 \ 0 & 1\end{array}\right] \)
Any matrix multiplied by its inverse results in the identity matrix, which is a way to verify the correctness of an inverse. For instance, if \(A\) is a matrix, and \(A^{-1}\) is its inverse, then:
\( AA^{-1} = I \)
and
\( A^{-1}A = I \).
Matrix Multiplication
Matrix multiplication involves taking the dot product of rows and columns from two matrices. For matrix \(A = \left[\begin{array}{rr}a & b \ c & d\end{array}\right]\) and matrix \(B = \left[\begin{array}{rr}e & f \ g & h\end{array}\right]\), their product is computed as:
  • The element at row 1, column 1: \( (a \cdot e + b \cdot g) \)
  • The element at row 1, column 2: \( (a \cdot f + b \cdot h) \)
  • The element at row 2, column 1: \( (c \cdot e + d \cdot g) \)
  • The element at row 2, column 2: \( (c \cdot f + d \cdot h) \)
In the provided solution, verifying the inverse involves multiplying \(A\) and \(A^{-1}\), resulting in the identity matrix.
Inverse Matrix
The inverse of a matrix is another matrix that, when multiplied with the original matrix, yields the identity matrix. For a 2x2 matrix \( \left[\begin{array}{rr}a & b \ c & d\end{array}\right] \), the inverse is given by:

\( A^{-1} = \frac{1}{\text{det}(A)} \left[\begin{array}{rr}d & -b \ -c & a\end{array}\right] \).
Given the matrix \(A = \left[\begin{array}{rr}4 & 3 \ -3 & -2\end{array}\right] \), we find its inverse as follows:

The determinant is 1, so the inverse matrix is simplified to:

\( A^{-1} = \left[\begin{array}{rr}-2 & -3 \ 3 & 4\end{array}\right] \).
Finally, we verify the correctness by ensuring both \(AA^{-1} = I\) and \(A^{-1}A = I\).

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Most popular questions from this chapter

Prove each of the following statements for any \(3 \times 3\) matrix \(A\). If \(A\) has two identical rows (or columns), then \(|A|=0\).

Solve each system of equations by using \(A^{-1} .\) Note that the matrix of coefficients in each system is a matrix $$ \begin{array}{l} x+6 y=4 \\ x+9 y=5 \end{array} $$

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) Determine whether each of the following statements is true, and explain your answer. \(A B=B A\) (commutative)

Solve each problem, using Cramer's rule and a graphing calculator. Gasoline Sales The Runway Deli sells regular unleaded, plus unleaded, and supreme unleaded Shell gasoline. The number of gallons of each grade and the total receipts for gasoline are shown in the table for the first three weeks of February. What was the price per gallon for each grade? $$\begin{array}{|c|c|c|c|c|} \hline & \text { Regular } & \text { Plus } & \text { Supreme } & \text { Receipts } \\ \hline \text { Week 1 } & 1270 & 980 & 890 & \$ 12,204.86 \\ \text { Week 2 } & 1450 & 1280 & 1050 & \$ 14,698.22 \\ \text { Week 3 } & 1340 & 1190 & 1060 & \$ 13,969.41 \\ \hline \end{array}$$

Write each matrix equation as a system of equations and solve the system by the method of your choice. $$\left[\begin{array}{ll}2 & 3 \\\4 & 6\end{array}\right]\left[\begin{array}{l}x \\\y\end{array}\right]=\left[\begin{array}{l}5 \\ 9\end{array}\right]$$
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