Question

Matrices \(A\) and \(B\) are given. Solve the matrix equation \(A X=B\). $$ \begin{array}{l} A=\left[\begin{array}{cc} 4 & -1 \\ -7 & 5 \end{array}\right] \\ B=\left[\begin{array}{cc} 8 & -31 \\ -27 & 38 \end{array}\right] \end{array} $$

Short answer

The solution matrix \( X \) is \( \begin{bmatrix} 1 & -9 \\ -5 & 1 \end{bmatrix} \).

Step-by-step solution

Check Invertibility of Matrix A

To solve the equation \( A X = B \), we need to ensure that matrix \( A \) is invertible, which allows us to multiply both sides by \( A^{-1} \). To determine if \( A \) is invertible, calculate the determinant \( \det(A) \). For matrix \( A = \begin{bmatrix} 4 & -1 \ -7 & 5 \end{bmatrix} \), the determinant is \( \det(A) = 4 \times 5 - (-1) \times (-7) = 20 - 7 = 13 \). Since the determinant is not zero, \( A \) is invertible.

Calculate the Inverse of Matrix A

The inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). For matrix \( A = \begin{bmatrix} 4 & -1 \ -7 & 5 \end{bmatrix} \), the inverse is \( A^{-1} = \frac{1}{13} \begin{bmatrix} 5 & 1 \ 7 & 4 \end{bmatrix} \), as already determined that \( ad-bc = 13 \).

Multiply Both Sides by the Inverse of A

To solve for \( X \), we compute \( X = A^{-1} B \). Use matrix multiplication to find \( X \). Multiplying \( A^{-1} = \frac{1}{13} \begin{bmatrix} 5 & 1 \ 7 & 4 \end{bmatrix} \) with \( B = \begin{bmatrix} 8 & -31 \ -27 & 38 \end{bmatrix} \), compute each element of \( X \).

Compute the Matrix Multiplication Result

Perform the multiplication: \[\left( \frac{1}{13} \begin{bmatrix} 5 & 1 \ 7 & 4 \end{bmatrix} \right)\begin{bmatrix} 8 & -31 \ -27 & 38 \end{bmatrix}\]Calculate each entry of the resulting matrix \( X \).Element \((1,1): \frac{1}{13} (5 \times 8 + 1 \times -27) = \frac{13}{13} = 1\).Element \((1,2): \frac{1}{13} (5 \times -31 + 1 \times 38) = \frac{-117}{13} = -9\).Element \((2,1): \frac{1}{13} (7 \times 8 + 4 \times -27) = \frac{-68}{13} = -5\).Element \((2,2): \frac{1}{13} (7 \times -31 + 4 \times 38) = \frac{13}{13} = 1\).Thus, \( X = \begin{bmatrix} 1 & -9 \ -5 & 1 \end{bmatrix} \).

Key concepts

Matrix inversion

Matrix inversion is a crucial concept in linear algebra. It helps solve matrix equations like \( AX = B \). To "invert" a matrix means to find another matrix that, when multiplied with the original one, gives the identity matrix. For instance, given a matrix \( A \), its inverse is denoted as \( A^{-1} \), and satisfies \( A \cdot A^{-1} = I \), where \( I \) is the identity matrix.

Only square matrices (matrices with the same number of rows and columns) potentially have inverses. The process involves complex algebraic manipulation and, typically, either row reduction methods or formulae applicable for specific square matrices, such as 2x2 matrices. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is calculated using the formula:

• \( A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \).

Being able to determine and compute matrix inverses is helpful in various fields, like engineering, physics, and computer graphics.

Determinant calculation

Determinants are scalar values associated with square matrices. They play a key role in understanding matrix properties. Calculating the determinant of a matrix helps us determine if the matrix is invertible.

For a 2x2 matrix, such as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \det(A) \) is given by the formula:

• \( ad - bc \)

If the determinant equals zero, the matrix is said to be singular, implying it does not have an inverse. A non-zero determinant indicates that the matrix is invertible.

In practical applications, determinants can reveal a lot about the geometry of linear transformations associated with the matrix, such as scaling effects and rotations.

Matrix multiplication

Matrix multiplication differs from regular arithmetic multiplication but it's just as useful in solving equations and performing transformations. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second.

When multiplying matrices, each entry in the resulting matrix is obtained by computing the dot product of corresponding rows and columns from the original matrices. For matrices \( A \text{ and } B \) given as: \[ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \] The product \( AB \) is a matrix where,

• Entry (1,1) = \( a_{11}b_{11} + a_{12}b_{21} \)
• Entry (1,2) = \( a_{11}b_{12} + a_{12}b_{22} \)
• Entry (2,1) = \( a_{21}b_{11} + a_{22}b_{21} \)
• Entry (2,2) = \( a_{21}b_{12} + a_{22}b_{22} \)

Understanding matrix multiplication is fundamental for transformations in higher dimensions, and is widely applied in computer graphics and systems modeling.

Invertibility of matrices

The invertibility of matrices is a pivotal aspect in linear algebra, determining whether a linear system has unique solutions. A matrix, \( A \), is invertible if there exists another matrix, \( A^{-1} \), such that their multiplication yields an identity matrix.

To determine if a matrix is invertible, calculating the determinant is critical. If \( \det(A) eq 0 \), the matrix is invertible. For our example with \( A = \begin{bmatrix} 4 & -1 \ -7 & 5 \end{bmatrix} \), the determinant was found to be 13, indicating invertibility.

Knowing whether a matrix can be inverted is essential because if it is non-invertible (singular), solving equations like \( AX = B \) becomes more complex since standard inversion methods fail. Applications of invertible matrices include solving systems of linear equations, cryptography, and computer graphics.