Question

Matrices \(A\) and \(B\) are given. Solve the matrix equation \(A X=B\). $$ \begin{array}{l} A=\left[\begin{array}{cc} 1 & -3 \\ -3 & 6 \end{array}\right] \\ B=\left[\begin{array}{cc} 12 & -10 \\ -27 & 27 \end{array}\right] \end{array} $$

Short answer

Matrix \(X\) is \(\begin{bmatrix} 30 & -64 \\ 18 & -27 \end{bmatrix}\).

Step-by-step solution

Determine if Matrix A is Invertible

To solve the matrix equation \(AX = B\), we need to check if matrix \(A\) is invertible. A matrix is invertible if its determinant is non-zero. Compute \(\text{det}(A)\):\[ \text{det}(A) = (1)(6) - (-3)(-3) = 6 - 9 = -3 \]. Since \(\text{det}(A) eq 0\), matrix \(A\) is invertible.

Calculate the Inverse of Matrix A

Since \(A\) is invertible, calculate its inverse. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the inverse is given by \(\frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\). Substitute the values from \(A\):\[ A^{-1} = \frac{1}{-3}\begin{bmatrix} 6 & 3 \ 3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & -1 \ -1 & -\frac{1}{3} \end{bmatrix} \].

Solve for Matrix X

Now we solve for \(X\) using the formula \(X = A^{-1}B\). Compute the product of \(A^{-1}\) and \(B\):\[ X = \begin{bmatrix} -2 & -1 \ -1 & -\frac{1}{3} \end{bmatrix}\begin{bmatrix} 12 & -10 \ -27 & 27 \end{bmatrix} = \begin{bmatrix} 30 & -64 \ 18 & -27 \end{bmatrix} \].

Verify the Solution

Check by substituting \(X\) back into the original equation \(AX = B\) to verify the solution. Compute \(AX\):\[ A\begin{bmatrix} 30 & -64 \ 18 & -27 \end{bmatrix} = \begin{bmatrix} 12 & -10 \ -27 & 27 \end{bmatrix} \]. Since the product \(AX\) equals \(B\), the solution is verified.

Key concepts

Matrix Inversion

Matrix inversion is a key concept in solving matrix equations. When you have an equation like \( AX = B \), inverting matrix \( A \) allows you to isolate \( X \), thus solving the equation. In simple terms, the inverse of a matrix \( A \), denoted \( A^{-1} \), is the matrix that when multiplied with \( A \) yields the identity matrix. The identity matrix acts like "1" does in regular multiplication. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is computed as \( \frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Remember, not all matrices have an inverse; Only those with a non-zero determinant do. Inverting a matrix is crucial for solving linear equations, transforming coordinates, and more.

Determinant

The determinant is a special number that can be calculated from a square matrix. It provides valuable information about the matrix, especially whether it is invertible. Calculating the determinant is straightforward for small matrices. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is \( ad - bc \). If the determinant is zero, the matrix does not have an inverse, which means you cannot solve the equation \( AX = B \) through inversion. A non-zero determinant tells you the matrix is invertible. Determinants are used in various areas of linear algebra, including solving systems of linear equations and finding eigenvalues.

Matrix Multiplication

Multiplying matrices is a fundamental operation in linear algebra, but it differs from multiplication of regular numbers. When multiplying two matrices \( A \) and \( B \), the number of columns in \( A \) must match the number of rows in \( B \). The resulting matrix will have the dimensions of the number of rows of \( A \) and the number of columns of \( B \). For each element in the resulting matrix, you multiply corresponding elements from rows of \( A \) by columns of \( B \) and sum them up. This operation is extensively used in solving equations, transformations, computer graphics, and more. It's important to follow the order, as matrix multiplication is not commutative, meaning \( AB \) does not necessarily equal \( BA \).

Linear Algebra

Linear algebra is a branch of mathematics revolving around vectors, vector spaces, and linear mappings. It is the foundation for handling systems of linear equations through methods like matrix inversion and determinant calculation. In this field, problems involving unknowns, like solving \( AX = B \), are commonly tackled. Linear algebra is widespread in applications such as computer science, engineering, physics, and economics, providing tools for modeling, analysis, and computation. Beyond just numbers, it deals with geometric concepts, transformations, and the structure of mathematical spaces, making it an essential subject in both theoretical and applied sciences.