(a) Write the following system as a matrix equation A X=B System Matrix equation 5 x+

step 1 For part A, we can write the following system in the form of a matrix equation such that A times

(a) Write the following system as a matrix equation A X=B ...

(a) Write the following system as a matrix equation $A X=B$ $$ \text { System } \quad \text { Matrix equation } $$ $$ \begin{array}{l}{5 x+3 y=4} \quad {A \quad \cdot \quad X \quad=\quad B}\\ {3 x+2 y=3}\end{array} $$ $\quad\quad\quad\quad\quad\quad[--]\quad[--]\quad=[--]$ (b) The inverse of $A$ is $A^{-1}=$ [_____] (c) The solution of the matrix equation is $X=A^{-1} B$ $$ X=\quad A^{-1} \quad B $$ $$ \left[\begin{array}{l}{x} \\ {y}\end{array}\right]={[--]\quad[--]=[--]} $$ (d) The solution of the system is $x=$ ______, $y=$ _______.

(a) Write the following system as a matrix equation $A X=B$
$$
\text { System } \quad \text { Matrix equation }
$$
$$
\begin{array}{l}{5 x+3 y=4} \quad {A \quad \cdot \quad X \quad=\quad B}\\ {3 x+2 y=3}\end{array}
$$
$\quad\quad\quad\quad\quad\quad[--]\quad[--]\quad=[--]$
(b) The inverse of $A$ is $A^{-1}=$ [_____]
(c) The solution of the matrix equation is $X=A^{-1} B$
$$
X=\quad A^{-1} \quad B
$$
$$
\left[\begin{array}{l}{x} \\ {y}\end{array}\right]={[--]\quad[--]=[--]}
$$
(d) The solution of the system is $x=$ ______, $y=$ _______.

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Key Concepts

Inverse of a Matrix

An inverse matrix, denoted A?¹, is a matrix such that when multiplied by the original matrix A, it yields the identity matrix (A?¹A = I). The concept of a matrix inverse is central in solving systems of linear equations, since if A is invertible, the solution can be found via multiplying both sides of the equation by A?¹. For a 2x2 matrix, the inverse has a specific formula using its determinant and swapped/coefficient sign adjustments.

Matrix Representation of Linear Systems

This concept involves expressing a system of linear equations in matrix form. The coefficients from the equations form a matrix (commonly denoted as A), the variables are organized into a column vector (denoted as X), and the constants from the right-hand side form another column vector (denoted as B). Writing the system as AX = B facilitates the use of matrix operations for solving the system.

Solving Linear Systems via Matrix Inversion

When a system of linear equations is represented as AX = B and the matrix A is invertible, the system can be solved succinctly by computing X = A?¹B. This approach leverages the properties of matrix multiplication and inverses, streamlining the process of finding the values of the variables that satisfy all the equations simultaneously.

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