Problem 52 - Advanced Engineering Mathematics

Question

Write the system in the form \(\mathbf{A} \mathbf{X}=\mathbf{B}\). Use \(\mathbf{X}=\mathbf{A}^{-1} \mathbf{B}\) to solve the system for each matrix \(\mathbf{B}\). $$ \begin{aligned} x_{1}+2 x_{2}+5 x_{3} &=b_{1} \\ 2 x_{1}+3 x_{2}+8 x_{3} &=b_{2} \\ -x_{1}+x_{2}+2 x_{3} &=b_{3} \\ \mathbf{B}=\left(\begin{array}{r} -1 \\ 4 \\ 6 \end{array}\right), \quad \mathbf{B} &=\left(\begin{array}{l} 3 \\ 3 \\ 3 \end{array}\right), & \mathbf{B}=\left(\begin{array}{r} 0 \\ -5 \\ 4 \end{array}\right) \end{aligned} $$

Step-by-Step Solution

Answer

For \(\mathbf{B}_1\), \(\mathbf{X} = \begin{pmatrix} -11 \\ 14 \\ -3 \end{pmatrix}\). For \(\mathbf{B}_2\), \(\mathbf{X} = \begin{pmatrix} 48 \\ -59 \\ 36 \end{pmatrix}\). For \(\mathbf{B}_3\), \(\mathbf{X} = \begin{pmatrix} -136 \\ 167 \\ -102 \end{pmatrix}\).

Step 1: Write the System as AX = B

To write the system in matrix form \(\mathbf{AX} = \mathbf{B}\), identify the coefficient matrix \(\mathbf{A}\) and the variable vector \(\mathbf{X}\). The given system is:\[\begin{aligned} x_1 + 2x_2 + 5x_3 &= b_1, \ 2x_1 + 3x_2 + 8x_3 &= b_2, \ -x_1 + x_2 + 2x_3 &= b_3.\end{aligned}\]Thus, the coefficient matrix \(\mathbf{A}\) and vector \(\mathbf{X}\) are:\[\mathbf{A} = \begin{pmatrix} 1 & 2 & 5 \ 2 & 3 & 8 \ -1 & 1 & 2 \end{pmatrix}, \quad \mathbf{X} = \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix}.\]Given \(\mathbf{B}\) vectors are:\[\mathbf{B}_1 = \begin{pmatrix} -1 \ 4 \ 6 \end{pmatrix}, \quad \mathbf{B}_2 = \begin{pmatrix} 3 \ 3 \ 3 \end{pmatrix}, \quad \mathbf{B}_3 = \begin{pmatrix} 0 \ -5 \ 4 \end{pmatrix}.\]

Step 2: Calculate the Inverse of A

The inverse of matrix \(\mathbf{A}\), \(\mathbf{A}^{-1}\), is calculated using the formula for a 3x3 matrix inverse:\[\mathbf{A}^{-1} = \frac{1}{|\mathbf{A}|}\text{adj}(\mathbf{A}),\]where \(|\mathbf{A}|\) is the determinant of \(\mathbf{A}\) and \(\text{adj}(\mathbf{A})\) is the adjugate of \(\mathbf{A}\). After computation:\[\mathbf{A}^{-1} = \begin{pmatrix} -1 & 18 & -11 \ 2 & -23 & 14 \ 0 & 5 & -3 \end{pmatrix}.\]

Step 3: Solve for X using Each B

Using the equation \(\mathbf{X} = \mathbf{A}^{-1} \mathbf{B}\), solve for \(\mathbf{X}\) for each \(\mathbf{B}\) matrix:- For \(\mathbf{B}_1 = \begin{pmatrix} -1 \ 4 \ 6 \end{pmatrix}\): \[ \mathbf{X}_1 = \mathbf{A}^{-1}\mathbf{B}_1 = \begin{pmatrix} -11 \ 14 \ -3 \end{pmatrix}. \]- For \(\mathbf{B}_2 = \begin{pmatrix} 3 \ 3 \ 3 \end{pmatrix}\): \[ \mathbf{X}_2 = \mathbf{A}^{-1}\mathbf{B}_2 = \begin{pmatrix} 48 \ -59 \ 36 \end{pmatrix}. \]- For \(\mathbf{B}_3 = \begin{pmatrix} 0 \ -5 \ 4 \end{pmatrix}\): \[ \mathbf{X}_3 = \mathbf{A}^{-1}\mathbf{B}_3 = \begin{pmatrix} -136 \ 167 \ -102 \end{pmatrix}. \]

Key Concepts

Understanding Inverse Matrices

The inverse of a matrix is somewhat akin to the reciprocal of a number; it's what you multiply the original matrix by to get the identity matrix. For a matrix \( \mathbf{A} \), its inverse is denoted as \( \mathbf{A}^{-1} \). Not all matrices have inverses, but when they do, calculating the inverse helps in solving matrix equations. The process involves finding the determinant and the adjugate of \( \mathbf{A} \).
To find \( \mathbf{A}^{-1} \), use the formula:

• \( \mathbf{A}^{-1} = \frac{1}{|\mathbf{A}|} \text{adj}(\mathbf{A}) \)

Where \( |\mathbf{A}| \) is the determinant, and \( \text{adj}(\mathbf{A}) \) is the adjugate matrix.
Remember, the product \( \mathbf{A} \cdot \mathbf{A}^{-1} \) results in the identity matrix, \( \mathbf{I} \), which is analogous to "1" in number operations.

Solving Systems of Linear Equations

Systems of linear equations involve finding the values of variables that satisfy multiple linear equations simultaneously. When such systems are written in matrix form as \( \mathbf{AX} = \mathbf{B} \), each equation is represented by a row in matrix \( \mathbf{A} \) multiplied by column vector \( \mathbf{X} \), equating to the corresponding entry in the column vector \( \mathbf{B} \).
The matrix approach allows solving these equations efficiently, especially when dealing with larger systems. By using the inverse of the matrix \( \mathbf{A} \), you can directly solve for the unknown vector \( \mathbf{X} \).
To find \( \mathbf{X} \), the equation is rearranged to:

• \( \mathbf{X} = \mathbf{A}^{-1} \mathbf{B} \)

This method is particularly useful because it converts the procedure into straightforward matrix multiplications, avoiding the need to manually solve each equation one by one.

The Role of Determinants

Determinants are mathematical entities associated with square matrices. They provide fundamental properties of matrices and have practical uses in various calculations. The determinant of a 3x3 matrix gives a scalar value depicting certain characteristics of the matrix. In solving matrix equations, the determinant is crucial for calculating the inverse.
A non-zero determinant means that the matrix is invertible; a zero determinant indicates that the matrix does not have an inverse. This concept is crucial for determining whether a particular matrix equation \( \mathbf{AX} = \mathbf{B} \) is solvable using the inverse.
The calculation of the determinant \( |\mathbf{A}| \) requires determinant-formulas or expansion by minors. Its value impacts the inverse and solutions to systems of equations, hence why this small number plays a significant role in linear algebra applications.