Question

Matrices \(A\) and \(B\) are given. Solve the matrix equation \(A X=B\). $$ A=\left[\begin{array}{cc} -4 & 1 \\ -1 & -2 \end{array}\right] $$ $$ B=\left[\begin{array}{ccc} -2 & -10 & 19 \\ 13 & 2 & -2 \end{array}\right] $$

Short answer

The solution is \(X = \begin{bmatrix} -1 & 2 & -4 \\ -6 & -2 & 3 \end{bmatrix}\).

Step-by-step solution

Identify Matrix Dimensions

Matrix \(A\) is a \(2 \times 2\) matrix and matrix \(B\) is a \(2 \times 3\) matrix. To solve \(A X = B\), the unknown matrix \(X\) must have the same number of rows as \(A\) and the same number of columns as \(B\), hence \(X\) is a \(2 \times 3\) matrix.

Find the Inverse of Matrix A

To solve \(A X = B\), we need to use the inverse of \(A\), denoted as \(A^{-1}\), given that it exists. First, check if \(A\) is invertible by calculating its determinant: \(\det(A) = (-4)(-2) - (1)(-1) = 8 + 1 = 9\), which is non-zero. Therefore, \(A\) is invertible. Now, calculate \(A^{-1}\) using the formula for the inverse of a \(2 \times 2\) matrix: \[ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} a_{22} & -a_{12} \ -a_{21} & a_{11} \end{bmatrix} \] Substituting values, \(A^{-1} = \frac{1}{9} \begin{bmatrix} -2 & -1 \ 1 & -4 \end{bmatrix}\).

Compute Matrix X

Use \(A^{-1}\) to find \(X\) by multiplying \(A^{-1}\) with \(B\): \(X = A^{-1}B\). Compute the product: \[ X = \frac{1}{9} \begin{bmatrix} -2 & -1 \ 1 & -4 \end{bmatrix} \begin{bmatrix} -2 & -10 & 19 \ 13 & 2 & -2 \end{bmatrix} \].

Perform Matrix Multiplication

Carry out the matrix multiplication for each element of \(X\) using row by column operations: - First row, first column: \((-2)(-2) + (-1)(13) = 4 - 13 = -9\), - First row, second column: \((-2)(-10) + (-1)(2) = 20 - 2 = 18\), - First row, third column: \((-2)(19) + (-1)(-2) = -38 + 2 = -36\), - Second row, first column: \((1)(-2) + (-4)(13) = -2 - 52 = -54\), - Second row, second column: \((1)(-10) + (-4)(2) = -10 - 8 = -18\), - Second row, third column: \((1)(19) + (-4)(-2) = 19 + 8 = 27\). Hence, \(X = \frac{1}{9} \begin{bmatrix} -9 & 18 & -36 \ -54 & -18 & 27 \end{bmatrix}\).

Simplify Matrix X

Multiply each element of the computed matrix by \(\frac{1}{9}\) to obtain the final answer for \(X\): - First row: \(\left[ -1, 2, -4 \right]\), - Second row: \(\left[ -6, -2, 3 \right]\). Thus, the solution is \(X = \begin{bmatrix} -1 & 2 & -4 \ -6 & -2 & 3 \end{bmatrix}\).

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves taking two matrices and producing a third one. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. In the problem provided, we multiplied a 2x2 matrix, specifically the inverse of matrix A, by a 2x3 matrix B. Each element of the resulting matrix is calculated by taking the dot product of corresponding rows and columns from the matrices being multiplied. Here’s a simple outline to follow to perform matrix multiplication:

• Identify the number of rows and columns in the matrices involved.
• Make sure the inner dimensions match (e.g., the number of columns in the first matrix is equal to the number of rows in the second matrix).
• Compute the dot product for each element in the resulting matrix by summing up the products of elements from the row of the first matrix and the column of the second matrix.

For example, if you have matrix A as \(\begin{bmatrix}a&b \ c&d\end{bmatrix}\) and matrix B as \(\begin{bmatrix}e&f&g \ h&i&j\end{bmatrix}\), the element in the first row and the first column of the resulting matrix is computed as \(ae+bh\). This process is repeated for each row and column to populate the entire result matrix.

Inverse of a Matrix

The inverse of a matrix is a critical concept in solving matrix equations, such as the matrix equation \(AX = B\). For a matrix to have an inverse, it must be square, meaning it has the same number of rows and columns, and it must have a non-zero determinant. The inverse of a matrix A, denoted as \(A^{-1}\), is the matrix that when multiplied by A, yields the identity matrix, which is a matrix with 1s along its diagonal and 0s elsewhere.

• To compute the inverse of a 2x2 matrix, use the formula: \(A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} a_{22} & -a_{12} \ -a_{21} & a_{11} \end{bmatrix}\).
• First, find the determinant of the matrix A.
• If the determinant is zero, the matrix does not have an inverse.
• If the determinant is non-zero, substitute the values into the formula and simplify.

In our problem, the determinant was calculated to be 9, which meant the inverse exists. Using this determinant, we found our inverse matrix A as \[\frac{1}{9} \begin{bmatrix} -2 & -1 \ 1 & -4 \end{bmatrix}\]. The inverse is then used to solve the matrix equation by multiplying it with matrix B.

Determinant of a Matrix

The determinant is a scalar value that is calculated from a square matrix. It provides important properties about the matrix, such as whether a matrix is invertible. A non-zero determinant indicates the matrix has an inverse, while a zero determinant means it does not.

• For a 2x2 matrix, the formula for the determinant is given by: \(\det(A) = a_{11}a_{22} - a_{12}a_{21}\).
• This formula essentially calculates the area covered by the matrix vectors in a geometric sense.
• A non-zero determinant means that the vectors are linearly independent, allowing for inversion.

In the exercise, matrix A is \(\begin{bmatrix}-4 & 1 \ -1 & -2\end{bmatrix}\), and its determinant is calculated as \((-4)(-2) - (1)(-1) = 8 + 1 = 9\). This positive result confirms that A is invertible.

Matrix Dimensions

Understanding matrix dimensions is key to any operation involving matrices, such as addition, multiplication, and solving matrix equations. The dimension of a matrix refers to its size, specified by the number of its rows and columns, written as rows x columns.

• In the context of matrix multiplication, dimensions determine how matrices can be multiplied. For example, to multiply two matrices, the number of columns in the first must equal the number of rows in the second.
• When solving matrix equations like \(AX = B\), the resulting matrix X's dimensions are derived from the rows of A and columns of B. This ensures compatibility with both A and B.

In our exercise, matrix A is a 2x2 matrix and matrix B is a 2x3 matrix. So, the solution matrix X must be a 2x3 matrix to satisfy the equation \(AX = B\). Recognizing dimensions can help prevent calculation errors and ensure that operations are possible.