Question

Matrices \(A\) and \(B\) are given. Solve the matrix equation \(A X=B\). $$ \begin{array}{l} A=\left[\begin{array}{ll} 3 & 3 \\ 6 & 4 \end{array}\right] \\ B=\left[\begin{array}{ll} 15 & -39 \\ 16 & -66 \end{array}\right] \end{array} $$

Short answer

The solution matrix \(X\) is \[\begin{bmatrix} -2 & -7 \\ 7 & -6 \end{bmatrix}\].

Step-by-step solution

Check Matrix Dimensions

First, ensure that matrices are compatible for multiplication: if \(A\) is \(2 \times 2\), \(X\) must be \(2 \times 2\) to match the columns of \(B\). Both sides are \(2 \times 2\) matrices, thus the dimensions align for multiplication.

Find the Inverse of Matrix A

Calculate the inverse of matrix \(A\). To find the inverse of a \(2 \times 2\) matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), use: \[ A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \] For \(A = \begin{bmatrix} 3 & 3 \ 6 & 4 \end{bmatrix} \), \(ad-bc = 3 \times 4 - 3 \times 6 = -6\). So, \[ A^{-1} = \frac{1}{-6} \begin{bmatrix} 4 & -3 \ -6 & 3 \end{bmatrix} = \begin{bmatrix} -\frac{2}{3} & \frac{1}{2} \ 1 & -\frac{1}{2} \end{bmatrix} \]

Multiply Both Sides by the Inverse of A

Multiply the inverse of \(A\) to both sides of the equation to isolate \(X\): \[ A^{-1}AX = A^{-1}B \] Since \(A^{-1}A = I\) (the identity matrix), the equation becomes \(IX = A^{-1}B\). Simplify to find \(X = A^{-1}B\).

Calculate X

Calculate \(X\) by multiplying \(A^{-1}\) with \(B\): \[ A^{-1} = \begin{bmatrix} -\frac{2}{3} & \frac{1}{2} \ 1 & -\frac{1}{2} \end{bmatrix}, \quad B = \begin{bmatrix} 15 & -39 \ 16 & -66 \end{bmatrix} \]Perform the multiplication:\[(A^{-1}B)_{11} = (-\frac{2}{3} \times 15) + (\frac{1}{2} \times 16) = -10 + 8 = -2 \]\[(A^{-1}B)_{12} = (-\frac{2}{3} \times (-39)) + (\frac{1}{2} \times (-66)) = 26-33 = -7 \]\[(A^{-1}B)_{21} = (1 \times 15) + (-\frac{1}{2} \times 16) = 15 - 8 = 7 \]\[(A^{-1}B)_{22} = (1 \times -39) + (-\frac{1}{2} \times -66) = -39 + 33 = -6 \]Thus, \[ X = \begin{bmatrix} -2 & -7 \ 7 & -6 \end{bmatrix} \]

Verify the Solution

Verify if multiplying matrix \(A\) by the calculated matrix \(X\) results in matrix \(B\): Calculate \(AX\):\[(AX)_{11} = (3 \times -2) + (3 \times 7) = -6 + 21 = 15 \]\[(AX)_{12} = (3 \times -7) + (3 \times -6) = -21 - 18 = -39 \]\[(AX)_{21} = (6 \times -2) + (4 \times 7) = -12 + 28 = 16 \]\[(AX)_{22} = (6 \times -7) + (4 \times -6) = -42 - 24 = -66 \]Resulting matrix \(AX\) matches matrix \(B\), confirming the solution is correct.

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves taking two matrices and producing another matrix. This operation is not the same as multiplying two numbers: the order of multiplication matters. For matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. This is why matrix dimensions are important, as seen when checking compatibility in the original exercise.
Matrix multiplication is done by taking the dot product of the rows of the first matrix with the columns of the second matrix. If we have two matrices, let's call them \(A\) and \(B\), their product \(C = AB\) is formed by multiplying each element of a row of \(A\) with each element of a column of \(B\), and then summing up these products for each entry in \(C\).

• This operation is essential in solving matrix equations, where we often need to multiply matrices or vectors to arrive at the correct answer.
• You can visualize the resulting matrix as being calculated row-wise and column-wise simultaneously.

Matrix Inverse

The inverse of a matrix is particularly important because it allows us to solve matrix equations like \(AX = B\) for \(X\). When a matrix \(A\) is multiplied by its inverse \(A^{-1}\), the result is the identity matrix (\(I\)).
Not all matrices have inverses. A matrix must be square (the same number of rows and columns) and have a non-zero determinant to have an inverse. For a \(2 \times 2\) matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is computed as:
\[ A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]

• The determinant \(ad-bc\) must be non-zero for this formula to work.
• In the exercise, solving \(AX = B\) involved computing \(A^{-1}\) correctly before further operations.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors, vector spaces, and linear transformations. It is particularly useful for solving systems of linear equations using matrices and provides tools for dealing with spaces of dimensions higher than three.
Matrices in linear algebra help represent linear transformations. They act on vectors or other matrices to transform them, which is integral when dealing with real-world applications such as computer graphics or data analysis.
Important concepts include:

• Vectors – which can be rows or columns within matrices.
• Matrices – used to represent and solve systems of linear equations.
• Determinants and eigenvalues – help understand properties of matrices, like invertibility and transformation characteristics.

The core of the exercise aligns with solving linear equations using the matrix equation format, providing a clear example of linear algebra in action.

2x2 Matrix

A \(2 \times 2\) matrix is one of the simplest square matrices, consisting of two rows and two columns. Despite its simplicity, calculating with a \(2 \times 2\) matrix covers many fundamental features of linear algebra.
Operations such as finding the determinant or inverse are considerably easier with \(2 \times 2\) matrices compared to larger ones. For instance, when solving equations such as \(AX = B\) in the given exercise, understanding the properties of a \(2 \times 2\) matrix simplifies calculations and teaching.
To calculate a determinant of a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), use:
\[ det(A) = ad-bc \]

• A non-zero determinant indicates that the matrix is invertible.
• In \(2 \times 2\) matrices, being able to quickly compute its inverse or solve a system makes them highly applicable in various fields.

Identity Matrix

An identity matrix is a special square matrix that does not change any vector it multiplies. It is often denoted as \(I_n\), with \(n\) referring to its dimension (for a \(2 \times 2\), it would be \(I_2\)).
The identity matrix has ones on its main diagonal and zeros elsewhere:

• For a \(2 \times 2\) identity matrix, it's \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).
• The property of an identity matrix \(A\) such that \(IA = AI = A\) is crucial in solving matrix equations where isolating the variable matrix is necessary.

In the solution process of the exercise, the identity matrix plays a key role. When multiplying the inverse of matrix \(A\) by matrix \(A\), you obtain the identity matrix. This simplification helps isolate matrix \(X\) and solve \(AX = B\) effectively. Understanding the identity matrix is thus essential to performing these matrix operations correctly.