Problem 10
Question
Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr}-7 & 4 \\ 8 & -5\end{array}\right]$$
Step-by-Step Solution
Verified Answer
The inverse matrix is \(\begin{bmatrix} -\frac{5}{3} & -\frac{4}{3} \\ -\frac{8}{3} & -\frac{7}{3} \end{bmatrix}\).
1Step 1: Determine if the Inverse Exists
For a matrix to have an inverse, its determinant must be non-zero. Calculate the determinant for the matrix \(A = \begin{bmatrix} -7 & 4 \ 8 & -5 \end{bmatrix}\) using the formula for a 2x2 matrix determinant: \(\text{det}(A) = ad - bc\), where \(a = -7\), \(b = 4\), \(c = 8\), \(d = -5\). So, \(\text{det}(A) = (-7)(-5) - (4)(8) = 35 - 32 = 3\). Since the determinant is not zero, the inverse exists.
2Step 2: Use the Formula for a 2x2 Matrix Inverse
The inverse of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by \(\frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\). Plug in the numbers: \(a = -7\), \(b = 4\), \(c = 8\), \(d = -5\). Substitute these into the inverse formula to get \(\frac{1}{3}\begin{bmatrix} -5 & -4 \ -8 & -7 \end{bmatrix}\).
3Step 3: Calculate the Inverse Matrix
Multiply each element in the matrix \(\begin{bmatrix} -5 & -4 \ -8 & -7 \end{bmatrix}\) by \(\frac{1}{3}\), resulting in \(\begin{bmatrix} -\frac{5}{3} & -\frac{4}{3} \ -\frac{8}{3} & -\frac{7}{3} \end{bmatrix}\). This matrix is the inverse of the given matrix.
Key Concepts
Determinant Calculation2x2 MatrixMatrix Algebra
Determinant Calculation
Understanding the calculation of a determinant, especially for a simple 2x2 matrix, is crucial in linear algebra. The determinant is a scalar value that can tell us a lot about a matrix, such as whether it is invertible. For a 2x2 matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]the determinant is calculated using the formula: \[\text{det}(A) = ad - bc\]
Basically, you multiply the top-left element \(a\) with the bottom-right element \(d\) and subtract the product of the other diagonal \(b\) and \(c\).
For the matrix \[\begin{bmatrix} -7 & 4 \ 8 & -5 \end{bmatrix}\],the determinant is \[ (-7)(-5) - (4)(8) = 35 - 32 = 3 \].
Since the determinant is not zero, the matrix is invertible, meaning it has an inverse. Calculating the determinant quickly tells us if we're on the right track to find an inverse, making it a powerful tool in matrix algebra.
Basically, you multiply the top-left element \(a\) with the bottom-right element \(d\) and subtract the product of the other diagonal \(b\) and \(c\).
For the matrix \[\begin{bmatrix} -7 & 4 \ 8 & -5 \end{bmatrix}\],the determinant is \[ (-7)(-5) - (4)(8) = 35 - 32 = 3 \].
Since the determinant is not zero, the matrix is invertible, meaning it has an inverse. Calculating the determinant quickly tells us if we're on the right track to find an inverse, making it a powerful tool in matrix algebra.
2x2 Matrix
A 2x2 matrix is the most basic square matrix in linear algebra and has the form:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]Two rows and two columns make it straightforward but still powerful for many applications. Understanding the operations on 2x2 matrices can serve as a foundation for tackling larger matrices.
In context, every element in a 2x2 matrix has a specific role, especially when calculating things like determinants or inverses. These matrices have properties that extend to larger matrices, and learning the fundamentals here helps build the skills necessary for advancing in topics like linear transformations or systems of equations.
In context, every element in a 2x2 matrix has a specific role, especially when calculating things like determinants or inverses. These matrices have properties that extend to larger matrices, and learning the fundamentals here helps build the skills necessary for advancing in topics like linear transformations or systems of equations.
- Easy to work with and compute by hand
- Essential for understanding basic matrix operations
- Forms the building block for more complex matrices
Matrix Algebra
Matrix algebra is a branch of mathematics centered around matrices, which are rectangular arrays of numbers. Operations with matrices are fundamental for solving systems of linear equations, transformations, and more.
Key operations include addition, subtraction, multiplication, and finding inverses. The process learned here, specifically finding the inverse, involves matrices with non-zero determinants. An inverse matrix essentially "undoes" the multiplication by the original matrix.
In algebra terms, if a matrix \(A\) has an inverse \(A^{-1}\), then \(A \cdot A^{-1} = I\), where \(I\) is the identity matrix.
Key operations include addition, subtraction, multiplication, and finding inverses. The process learned here, specifically finding the inverse, involves matrices with non-zero determinants. An inverse matrix essentially "undoes" the multiplication by the original matrix.
In algebra terms, if a matrix \(A\) has an inverse \(A^{-1}\), then \(A \cdot A^{-1} = I\), where \(I\) is the identity matrix.
- Inverse matrices help solve linear equations
- Understanding matrices allows for transformations and coordinate changes
- Crucial for fields like computer graphics, quantum physics, and economics
Other exercises in this chapter
Problem 10
Use back-substitution to solve the triangular system. $$\left\\{\begin{aligned} 4 x\quad\quad+3 z &=10 \\ 2 y-\quad z &=-6 \\ \frac{1}{2} z &=4 \end{aligned}\ri
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Graph the inequality. $$-x^{2}+y \geq 10$$
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Perform the matrix operation, or if it is impossible, explain why. $$\left[\begin{array}{rr} 2 & -3 \\ 0 & 1 \\ 1 & 2 \end{array}\right]\left[\begin{array}{l} 5
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A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the syste
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