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Magazine Chapter 12: Problem 15
For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{ll} -2 & 5 \\ -3 & 6 \end{array}\right] $$
Short Answer
Expert verified
The inverse is \( \left[ \begin{array}{cc} 2 & -\frac{5}{3} \\ 1 & -\frac{2}{3} \end{array} \right] \).
Step by step solution
01
Identify the matrix
First, we examine the given matrix, which is a 2x2 matrix: \( A = \left[ \begin{array}{ll} -2 & 5 \ -3 & 6 \end{array} \right] \). We need to find the inverse, if it exists.
02
Calculate the determinant
For a 2x2 matrix \( A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \), the determinant is given by \( \text{det}(A) = ad - bc \). Here, \( a = -2 \), \( b = 5 \), \( c = -3 \), and \( d = 6 \). Substitute these values into the determinant formula: \( \text{det}(A) = (-2)(6) - (5)(-3) = -12 + 15 = 3 \).
03
Determine if the inverse exists
A matrix has an inverse if and only if its determinant is non-zero. Since \( \text{det}(A) = 3 \), which is not zero, the matrix does have an inverse.
04
Find the inverse matrix
For a 2x2 matrix \( A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) with determinant \( \text{det}(A) eq 0 \), the inverse is computed as \( A^{-1} = \frac{1}{\text{det}(A)} \left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right] \). For our matrix, this becomes: \[ A^{-1} = \frac{1}{3} \left[ \begin{array}{cc} 6 & -5 \ 3 & -2 \end{array} \right] = \left[ \begin{array}{cc} 2 & -\frac{5}{3} \ 1 & -\frac{2}{3} \end{array} \right] \]
05
Verify the inverse
To confirm that we have calculated the inverse correctly, we multiply the original matrix by its inverse. \( A \times A^{-1} = \left[ \begin{array}{cc} -2 & 5 \ -3 & 6 \end{array} \right] \times \left[ \begin{array}{cc} 2 & -\frac{5}{3} \ 1 & -\frac{2}{3} \end{array} \right] \). After performing the multiplication, ensure that the result is the identity matrix \( I = \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \). This confirms the inverse is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Determinant Calculation
Calculating the determinant is an essential step in finding the inverse of a matrix. For a 2x2 matrix, the determinant provides a single number that summarizes certain properties of the matrix. It is computed with the formula:
- For a matrix \( A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \), the determinant is \( \text{det}(A) = ad - bc \).
- To find if a matrix has an inverse, its determinant must not be zero.
- \( a = -2, \ b = 5, \ c = -3, \ d = 6 \)
- The determinant is: \( (-2)(6) - (5)(-3) = -12 + 15 = 3 \).
2x2 Matrix
A 2x2 matrix is one of the most straightforward matrices in linear algebra, consisting of two rows and two columns. Its compactness makes it a perfect candidate for beginners learning about matrix operations. Here鈥檚 what you need to know:
- A 2x2 matrix is represented as \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \).
- Each element, such as \( a \), \( b \), \( c \), and \( d \), impacts the matrix鈥檚 properties, like its determinant.
- 2x2 matrices are much easier to work with compared to larger matrices, making operations like finding determinants and inverses simpler.
Identity Matrix
The identity matrix plays a crucial role in matrix multiplication, particularly when discussing matrix inverses. It acts similarly to the number 1 in multiplication of real numbers. Key points include:
- An identity matrix for a 2x2 matrix is \( I = \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \).
- When any matrix is multiplied by its inverse, the result should be the identity matrix.
- This property confirms that the matrices are true inverses of each other.
Matrix Multiplication
Matrix multiplication involves a systematic way of combining two matrices to produce a new matrix. It's essential for verifying the inverse of matrices. Here鈥檚 how it works:
- For two matrices \( A \) and \( B \), multiply each element of a row from \( A \) by each element of a column from \( B \), summing the products.
- The product matrix should be of dimensions corresponding to the outer dimensions of the multiplied matrices (e.g., a 2x2 multiplied by a 2x2 results in another 2x2 matrix).
- In our example, multiply \( \left[ \begin{array}{cc} -2 & 5 \ -3 & 6 \end{array} \right] \) by its inverse to ensure the result is \( \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \).
