Given the square matrix [tex]A=\left(\begin{array}{ccc}1 & 2 & -3 \\ 2 & -1 & -1 \\ 3 & 2 & 1\end{array}\right)[/tex], find [tex]A^{-1}[/tex], the inverse of matrix [tex]A[/tex], and use your result to solve the following system of linear equations:
[tex]\begin{array}{l}
x+2 y-3 z=3 \\
2 x-y-z=11 \\
3 x+2 y+z=-5
\end{array}[/tex]
Define the following with an example each:
(i) Null matrix
(ii) Square matrix
(iii) Symmetric Matrix
Answer (2)
Calculate the determinant of matrix A: ∣ A ∣ = − 30 .
Find the inverse of matrix A: A − 1 = − 30 1 6 1 − 30 7 15 4 − 3 1 − 15 2 6 1 6 1 6 1 .
Solve the system of equations A X = B using X = A − 1 B : X = 2 − 4 − 3 .
The solution to the system of equations is: 2 − 4 − 3 .
Explanation
Problem Analysis First, we analyze the given information. We have a matrix A and a system of linear equations. Our goal is to find the inverse of A , use it to solve the system of equations, and define Null, Square, and Symmetric matrices with examples.
Finding the Inverse of A We are given the matrix A = 1 2 3 2 − 1 2 − 3 − 1 1 We need to find its inverse A − 1 .
Determinant of A Using a calculator, the determinant of A is calculated as: ∣ A ∣ = − 30 Since the determinant is non-zero, the inverse exists.
Calculating the Inverse Matrix The inverse of A is calculated as: A − 1 = − 0.0333 0.1667 − 0.2333 0.2667 − 0.3333 − 0.1333 0.1667 0.1667 0.1667 = − 30 1 6 1 − 30 7 15 4 − 3 1 − 15 2 6 1 6 1 6 1
System of Equations in Matrix Form Now we have the system of equations: ⎩ ⎨ ⎧ x + 2 y − 3 z = 3 2 x − y − z = 11 3 x + 2 y + z = − 5 We can write this in matrix form as A X = B , where X = x y z and B = 3 11 − 5 .
Solving for X To solve for X , we use the formula X = A − 1 B . Thus, X = − 30 1 6 1 − 30 7 15 4 − 3 1 − 15 2 6 1 6 1 6 1 3 11 − 5 = 2 − 4 − 3 So, x = 2 , y = − 4 , and z = − 3 .
Definitions and Examples (i) Null Matrix: A matrix in which all the elements are zero. For example: ( 0 0 0 0 ) (ii) Square Matrix: A matrix with the same number of rows and columns. For example: ( 1 3 2 4 ) (iii) Symmetric Matrix: A square matrix that is equal to its transpose. For example: ( 1 2 2 3 )
Final Answer Therefore, the inverse of matrix A is A − 1 = − 30 1 6 1 − 30 7 15 4 − 3 1 − 15 2 6 1 6 1 6 1 and the solution to the system of equations is x = 2 , y = − 4 , z = − 3 .
Examples
In engineering, solving systems of linear equations is crucial for analyzing circuits, determining structural stability, and optimizing control systems. For instance, when designing a bridge, engineers use systems of equations to calculate the forces and stresses acting on different parts of the structure. The matrix A represents the coefficients of these forces, and solving the system provides the values of the unknown forces, ensuring the bridge's stability and safety. Similarly, in economics, these methods can model supply and demand to find market equilibrium.
To find the inverse of matrix A , we computed its determinant and then calculated the adjugate to derive A − 1 . We solved the provided system of equations using the inverse, yielding the solution x = 2 , y = − 4 , and z = − 3 . Additionally, definitions of null, square, and symmetric matrices were provided with examples for clarity.
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