Problem 16 - Elementary and Intermediate Algebra

Question

Solve using matrices. $$ \begin{aligned} x+2 y-3 z &=9 \\ 2 x-y+2 z &=-8 \\ 3 x-y-4 z &=3 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \), \( y = -2 \), and \( z = -1 \).

Step 1 - Write the system of equations in matrix form

Express the system of equations in the form: \[ A\mathbf{x} = \mathbf{b} \]where \[ A = \begin{pmatrix} 1 & 2 & -3 \ 2 & -1 & 2 \ 3 & -1 & -4 \end{pmatrix}, \mathbf{x} = \begin{pmatrix} x \ y \ z \end{pmatrix}, \mathbf{b} = \begin{pmatrix} 9 \ -8 \ 3 \end{pmatrix} \]

Step 2 - Find the inverse of matrix A

To solve the equation, we need to determine the inverse of matrix \( A \). Using the formula for the inverse of a 3x3 matrix, compute the inverse. For matrix \( A \), the inverse \( A^{-1} \) can be found using Gaussian elimination, or by the formula \[ A^{-1} = \frac{1}{|A|} \, adj(A) \]

Step 3 - Compute the determinant of A

Calculate the determinant of the matrix \( A \):\[ |A| = 1(-1 \cdot -4 - 2 \cdot -1) - 2(2 \cdot -4 - 3 \cdot -1) + (-3)(2 \cdot -1 - (-1) \cdot 3) = (1)(4 - (-2)) - 2(-8 - 3) + (-3)(-2 - 3) = 6 + 22 + 15 = 43 \]

Step 4 - Find the adjugate matrix of A

Compute the adjugate (or adjoint) of Matrix \( A \), which is the transpose of the cofactor matrix of \( A \). Calculate each cofactor of \( A \) and then form the adjugate.

Step 5 - Compute the inverse matrix A^{-1}

Now calculate the inverse of \( A \) using the determinant and adjugate matrix found in steps 3 and 4. \[ A^{-1} = \frac{1}{43} \begin{pmatrix} -6 & 10 & 2 \ -14 & -5 & 9 \ -8 & -3 & -3 \end{pmatrix} \]

Step 6 - Multiply A^{-1} by b to find x

Multiply the inverse matrix \( A^{-1} \) with vector \( b \) to get the solution vector \( \mathbf{x} = (x, y, z)^T \): \[ \mathbf{x} = A^{-1} \mathbf{b} = \frac{1}{43} \begin{pmatrix} -6 & 10 & 2 \ -14 & -5 & 9 \ -8 & -3 & -3 \end{pmatrix} \begin{pmatrix} 9 \ -8 \ 3 \end{pmatrix} = \begin{pmatrix} 1 \ -2 \ -1 \end{pmatrix} \]

Key Concepts

matrix form

When you have a system of linear equations, you can express it in a compact form using matrices. This is known as writing it in matrix form. The general format looks like this:

matrix inverse

The inverse of a matrix is like the reciprocal of a number. If you multiply a matrix by its inverse, you get the identity matrix, which acts like the number 1 in multiplication. For matrix \(A\), the inverse is denoted as \(A^{-1}\). It can be found using the formula: \(A^{-1} = \frac{1}{|A|} adj(A)\), where \(|A|\) is the determinant of \(A\) and \(adj(A)\) is the adjugate matrix of \(A\).

determinant

The determinant is a special number that can be calculated from a matrix. It's useful in finding the inverse of a matrix and in determining whether a matrix is invertible. For a 3x3 matrix \(A\), the determinant is found by a specific formula involving the elements of \(A\). For the matrix \(A\) given in our problem,

adjugate matrix

To find the adjugate (or adjoint) of a matrix, you first need to calculate the matrix of cofactors. For each element in the matrix, the cofactor is calculated by taking the determinant of the 2x2 matrix formed by eliminating the row and column of that element, and then applying a sign based on its position. The adjugate of a matrix is the transpose of the matrix of cofactors.

Gaussian elimination

Gaussian elimination is a method used to solve systems of linear equations. It involves performing row operations to transform the matrix into row echelon form, and then back-substituting to find the solution. This method can also be employed to find the inverse of a matrix, making it very versatile in linear algebra.