Problem 41 - Advanced Engineering Mathematics

Question

$$ \text { Write }\left(\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right)\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right)=\left(\begin{array}{l} b_{1} \\ b_{2} \end{array}\right) \text { without matrices. } $$

Step-by-Step Solution

Answer

Transform matrix equation into system: \( a_{11}x_1 + a_{12}x_2 = b_1 \) and \( a_{21}x_1 + a_{22}x_2 = b_2 \).

Step 1: Define the Equation System

You have a matrix equation that represents a system of linear equations. The given matrix equation is:\[ \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \end{pmatrix} = \begin{pmatrix} b_1 \ b_2 \end{pmatrix} \]. This corresponds to the system of equations: \( a_{11}x_1 + a_{12}x_2 = b_1 \) and \( a_{21}x_1 + a_{22}x_2 = b_2 \).

Step 2: Substitute the Matrices into Equations

Substitute the elements of the matrix into two separate equations to eliminate the matrix notation. This will give you:1. Equation 1: \( a_{11}x_1 + a_{12}x_2 = b_1 \)2. Equation 2: \( a_{21}x_1 + a_{22}x_2 = b_2 \).

Step 3: Solve the System of Equations (Optional)

If necessary, solve the system of equations derived:1. Use either substitution or elimination method. For substitution, solve one equation for one variable and substitute into the other. For elimination, add or subtract the equations to eliminate a variable.2. Determine the values of \( x_1 \) and \( x_2 \). This step can provide specific solutions if the values of \( a_{11}, a_{12}, a_{21}, a_{22}, b_1, b_2 \) are known.

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In the context of linear algebra, these equations are typically linear. Each equation in a system describes a line, plane, or hyperplane, depending on the number of variables. The solution to the system is the point or set of points where the lines or planes intersect.

For example, consider a system of two equations:

• \( a_{11}x_1 + a_{12}x_2 = b_1 \)
• \( a_{21}x_1 + a_{22}x_2 = b_2 \)

Here, \( x_1 \) and \( x_2 \) are the variables we aim to solve for. These equations represent lines on a plane, and the solution is the intersection point of these lines.

Systems of equations can be solved using various methods, including substitution and elimination, which we will discuss in more detail later.

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra. It involves multiplying rows by columns. In the given exercise, matrix multiplication is used to turn matrix equations into systems of equations.

Consider the matrix equation:

• \( \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \end{pmatrix} = \begin{pmatrix} b_1 \ b_2 \end{pmatrix} \)

To multiply these matrices, take the dot product of each row of the first matrix with the column vector, resulting in a new column vector.

Multiply:

• First row: \( a_{11}x_1 + a_{12}x_2 \)
• Second row: \( a_{21}x_1 + a_{22}x_2 \)

This gives you a set of equations corresponding to each element of the resulting vector. Matrices allow us to efficiently express and solve systems of equations, especially large ones.

Substitution Method

The substitution method is a technique for solving systems of equations. It involves solving one equation for one variable and substituting that expression into the other equation. Using the substitution method effectively requires two main steps:

1. Solve one equation for one of the variables.2. Substitute the expression obtained into the other equation.

Let's say we solve the first equation for \( x_1 \):

• \( x_1 = \frac{b_1 - a_{12}x_2}{a_{11}} \)

Then substitute \( x_1 \) into the second equation:

• \( a_{21}\left(\frac{b_1 - a_{12}x_2}{a_{11}}\right) + a_{22}x_2 = b_2 \)

From here, solve for \( x_2 \). Once \( x_2 \) is known, plug it back into the expression found for \( x_1 \) to find its value.

This method is straightforward when the equations are simple or the coefficients allow easy manipulation.

Elimination Method

The elimination method is another technique for solving systems of equations. It involves adding or subtracting equations to eliminate one variable, allowing for the easy solution of the remaining variable.

Here's how you can approach elimination:

• Multiply each equation by a number that will make the coefficients of one of the variables equal.
• Add or subtract the equations, canceling out one of the variables.

For instance, consider the equations:

• \( a_{11}x_1 + a_{12}x_2 = b_1 \)
• \( a_{21}x_1 + a_{22}x_2 = b_2 \)

Multiply the first equation by \( a_{22} \) and the second by \( a_{12} \), then subtract to eliminate \( x_2 \):

• \( a_{22}(a_{11}x_1 + a_{12}x_2) - a_{12}(a_{21}x_1 + a_{22}x_2) = a_{22}b_1 - a_{12}b_2 \)

This leaves an equation with only \( x_1 \), which can be solved easily. Once \( x_1 \) is known, substitute back to find \( x_2 \).

The elimination method is particularly useful when dealing with fractions or complex coefficients, as it simplifies the solving process.