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Chapter 7: Problem 61

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{rr} 5 & -9 \\ -1 & 2 \end{array}\right]$$

Short Answer

Expert verified
The original matrix \(A\) is \(\begin{pmatrix} 2 & 9 \\ 1 & 5 \end{pmatrix}\).

Step by step solution

01

Understand the Problem

We are given the inverse of a square matrix, denoted as \(A^{-1}\), and we need to find the original matrix \(A\). The relationship between a square matrix and its inverse is that their product equals the identity matrix.
02

Inverse Matrix Formula

For a 2x2 matrix \(B\) given by \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the inverse, if it exists, is \(B^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}\). Since \(A^{-1}\) is given, we need to find a matrix \(A\) such that \(A \cdot A^{-1} = I\), where \(I\) is the identity matrix.
03

Verify the Structure of Matrix \(A\)

Assuming \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), \(A^{-1}\) should have the structure given by the inverse formula previously mentioned. Given \(A^{-1} = \begin{pmatrix} 5 & -9 \ -1 & 2 \end{pmatrix}\), using the formula implies the original matrix \(A\) should have its elements correspond as follows: \(a = 2\), \(b = 9\), \(c = 1\), \(d = 5\).
04

Equation Verification

We hypothesize \(A\) as \(\begin{pmatrix} 2 & 9 \ 1 & 5 \end{pmatrix}\). Verify by checking if \(A \cdot A^{-1} = I\), where \(I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\). Compute the multiplication to see if it results in the identity matrix.
05

Calculate the Product \(A \cdot A^{-1}\)

Multiply the matrices: \(\begin{pmatrix} 2 & 9 \ 1 & 5 \end{pmatrix} \times \begin{pmatrix} 5 & -9 \ -1 & 2 \end{pmatrix} = \begin{pmatrix} (2 \cdot 5) + (9 \cdot -1) & (2 \cdot -9) + (9 \cdot 2) \ (1 \cdot 5) + (5 \cdot -1) & (1 \cdot -9) + (5 \cdot 2) \end{pmatrix}\). This simplifies to \(\begin{pmatrix} 10 - 9 & -18 + 18 \ 5 - 5 & -9 + 10 \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\), confirming it is indeed the identity matrix.

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Key Concepts

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

Square Matrix
A square matrix is a fundamental concept in linear algebra where the number of rows and columns in the matrix are equal. This makes it an n x n matrix. These matrices are particularly significant because they often have unique properties compared to non-square matrices.

### Why Square Matrices Matter
  • They can potentially have inverses. A square matrix has an inverse if its determinant is not zero.
  • Square matrices can be added if they are of the same dimension.
  • They are essential in theories related to vector spaces, for example, eigenvalues and eigenvectors relate to square matrices.
### Real-World Example Suppose you have a 2x2 square matrix representing a transformation in computer graphics. Understanding how this matrix transforms a shape can help create desired graphical effects. Recognizing when a matrix is square is the first step in determining its possible applications in these complex operations.
Identity Matrix
The identity matrix is like the number 1 in matrix algebra. It is a special kind of square matrix that, when multiplied with any compatible matrix, returns the original matrix. This means the identity matrix does not change the matrix it multiplies.

### Properties of Identity Matrices
  • The identity matrix is always a square matrix.
  • Its diagonal elements are all ones, and all other elements are zeros. For example, a 2x2 identity matrix looks like \( \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \).
  • Multiplying any matrix by the identity matrix will return the original matrix, both from the left and the right.
### Importance in Matrix EquationsIf you are solving for a matrix inverse in practical applications like encryption algorithms or inverting transformations, the identity matrix often serves as your check figure. After finding a supposed inverse, multiplying it by the original matrix should yield the identity matrix, confirming the inverse is correct.
Matrix Multiplication
Matrix multiplication is a core operation in linear algebra. It involves combining two matrices to produce a third matrix. This operation, while not as straightforward as multiplication of numbers, follows specific rules.

### Performing Matrix Multiplication
  • The number of columns in the first matrix must equal the number of rows in the second.
  • To find an element in the resulting matrix, take the dot product of the corresponding row from the first matrix with the column from the second matrix.
  • This operation is not commutative, meaning \( A \times B eq B \times A \) in general.
### Application ExampleIf you have a matrix representing a geometric transformation and another representing data points, multiplying these can yield the transformed data points. Understanding matrix multiplication helps in fields like computer graphics, data science, and in solving systems of equations.

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Most popular questions from this chapter

Solve each system by using the matrix inverse method. $$\begin{aligned} x+3 y-2 z-w &=9 \\ 4 x+y+z+2 w &=2 \\ -3 x-y+z-w &=-5 \\ x-y-3 z-2 w &=2 \end{aligned}$$

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$y=2 x-1$$ $$y=2-x^{2}$$

Let \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right],\) where \(a, b,\) and \(c\) are nonzero real numbers. Find \(A^{-1}\).

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{rrr} \frac{2}{3} & -\frac{1}{3} & 0 \\ \frac{1}{3} & -\frac{5}{3} & 1 \\ \frac{1}{3} & \frac{1}{3} & 0 \end{array}\right]$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}x-y &<1 \\\\-1
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