Problem 39 - Precalculus Mathematics for Calculus

Question

Solve the matrix equation by multiplying each side by the appropriate inverse matrix. $$\left[\begin{array}{rr} 3 & -2 \\ -4 & 3 \end{array}\right]\left[\begin{array}{lll} x & y & z \\ u & v & w \end{array}\right]=\left[\begin{array}{llr} 1 & 0 & -1 \\ 2 & 1 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix \( B = \left[\begin{array}{lll} 7 & 2 & 3 \\ 10 & 3 & 5 \end{array}\right] \).

Step 1: Identify Matrix A

In this problem, Matrix A is the square matrix we need to invert. It is given as \( A = \left[\begin{array}{rr} 3 & -2 \ -4 & 3 \end{array}\right] \). We need to find the inverse of this matrix to further solve the equation.

Step 2: Determine the Inverse of Matrix A

The inverse of a 2x2 matrix \(A = \left[\begin{array}{rr} a & b \ c & d \end{array}\right] \) is given by \( A^{-1} = \frac{1}{ad-bc} \left[\begin{array}{rr} d & -b \ -c & a \end{array}\right] \), provided \(ad-bc eq 0\). Calculate the determinant: \( 3 \times 3 - (-2) \times (-4) = 9 - 8 = 1 \). The inverse is therefore \( A^{-1} = \left[\begin{array}{rr} 3 & 2 \ 4 & 3 \end{array}\right] \).

Step 3: Multiply Both Sides of the Equation by the Inverse

Multiply both sides of the matrix equation \(A\cdot B = C\) by \(A^{-1}\) on the left: \(A^{-1}A\cdot B = A^{-1}C\). This simplifies to \(B = A^{-1}C\), as \(A^{-1}A = I\) (the identity matrix).

Step 4: Compute Matrix B

Calculate the matrix product \( A^{-1} \left[\begin{array}{llr} 1 & 0 & -1 \ 2 & 1 & 3 \end{array}\right] \):Multiply row by column for each element of the resulting 2x3 matrix.1. First row, first column: \(3\times1 + 2\times2 = 7\)2. First row, second column: \(3\times0 + 2\times1 = 2\)3. First row, third column: \(3\times-1 + 2\times3 = 3\)4. Second row, first column: \(4\times1 + 3\times2 = 10\)5. Second row, second column: \(4\times0 + 3\times1 = 3\)6. Second row, third column: \(4\times-1 + 3\times3 = 5\)So, \( B = \left[\begin{array}{lll} 7 & 2 & 3 \ 10 & 3 & 5 \end{array}\right] \).

Step 5: Conclusion

After multiplying the inverse of matrix A with matrix C, we have solved for matrix B, which contains the unknowns: \( B = \left[\begin{array}{lll} 7 & 2 & 3 \ 10 & 3 & 5 \end{array}\right] \).

Key Concepts

Matrix Inversion

The concept of matrix inversion is fundamental when solving matrix equations. In simple terms, the inverse of a matrix is a matrix that, when multiplied with the original matrix, results in the identity matrix. The identity matrix is like the number 1 for regular numbers, meaning it doesn't change another matrix when used in multiplication.
To find the inverse of a 2x2 matrix, say \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), you need to calculate its determinant first. The determinant is given by \(ad - bc\).
If the determinant is not zero (since a zero determinant means the matrix has no inverse), the inverse of the matrix \(A\) is calculated by swapping the positions of \(a\) and \(d\), and changing the signs of \(b\) and \(c\), followed by dividing each term by the determinant. This results in the inverse \(A^{-1} = \begin{bmatrix} \frac{d}{ad-bc} & \frac{-b}{ad-bc} \ \frac{-c}{ad-bc} & \frac{a}{ad-bc} \end{bmatrix}\). Using these steps, you can find the inverse of a matrix and use it in solving matrix equations.
For example, with matrix \(A\) in the problem, the determinant is 1, resulting in \(A^{-1} = \begin{bmatrix} 3 & 2 \ 4 & 3 \end{bmatrix}\).

Determinants

Determinants offer a way to determine certain properties of matrices. Most importantly, the determinant helps ascertain whether a matrix is invertible. If the determinant is zero, the matrix doesn't have an inverse; otherwise, it does.
For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is straightforward to calculate: \(ad - bc\).
In the exercise, the matrix \(A\) had a determinant equal to 1. This means \(A\) is invertible. Calculating the determinant allows us to ensure that the formula for the inverse can be applied safely.
Another practical aspect of determinants is that they can be used to understand geometric properties of linear transformations represented by matrices. Although this aspect may be beyond the current problem's scope, it is a key reason why determinants are so valuable in mathematics.

Matrix Multiplication

Matrix multiplication is a process that allows us to combine two matrices into a single new matrix. This operation is crucial for solving matrix equations such as the one in the original problem.
To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. In the case of multiplying a 2x2 matrix by a 2x3 matrix, the process involves summing the products of entries from the rows of the first matrix and the columns of the second matrix.
In our example, once we found the inverse of matrix \(A\), we multiplied it by matrix \(C\) on the right-hand side of the equation. Each element of the resulting matrix is computed by multiplying rows by columns: for a position \((i, j)\) in the result, we multiply each corresponding item in row \(i\) of the first matrix with column \(j\) of the second matrix, summing the results together.
This step-by-step approach resulted in matrix \(B = \begin{bmatrix} 7 & 2 & 3 \ 10 & 3 & 5 \end{bmatrix}\), effectively solving the matrix equation. Understanding how matrix multiplication works is essential for manipulating and solving equations involving matrices.