Textbook Question
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
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7. Systems of Equations & Matrices
Determinants and Cramer's Rule
4:19 minutesProblem 39
Textbook Question
a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.
Verified step by step guidance1
Step 1: Write the system of equations in matrix form \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the column matrix of variables, and \(B\) is the constants matrix. For the system:
\[\begin{cases} x - y + z = 8 \\ 2y - z = -7 \\ 2x + 3y = 1 \end{cases}\]
The coefficient matrix \(A\) is:
\[A = \begin{bmatrix} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \end{bmatrix}\]
The variable matrix \(X\) is:
\[X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}\]
The constants matrix \(B\) is:
\[B = \begin{bmatrix} 8 \\ -7 \\ 1 \end{bmatrix}\]
Step 2: Confirm the inverse matrix \(A^{-1}\) is given as:
\[A^{-1} = \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}\]
Step 3: Use the matrix equation \(X = A^{-1}B\) to find the solution vector \(X\). This means multiplying the inverse matrix \(A^{-1}\) by the constants matrix \(B\).
Step 4: Perform the matrix multiplication:
\[X = \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} \times \begin{bmatrix} 8 \\ -7 \\ 1 \end{bmatrix}\]
Multiply each row of \(A^{-1}\) by the column matrix \(B\) to get each variable \(x\), \(y\), and \(z\).
Step 5: Write the resulting matrix \(X\) as the solution to the system, where the first element is \(x\), the second is \(y\), and the third is \(z\).
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Video duration:
4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Matrix Representation of Linear Systems
A system of linear equations can be expressed as a matrix equation AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the constants matrix. This form simplifies solving the system using matrix operations.
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Introduction to Systems of Linear Equations
Matrix Inverse and Its Role in Solving Systems
The inverse of a matrix A, denoted A⁻¹, is used to solve AX = B by multiplying both sides by A⁻¹, yielding X = A⁻¹B. This method works only if A is invertible (non-singular), providing a direct way to find the solution vector X.
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Matrix Multiplication for Solution Computation
To find the solution vector X, multiply the inverse matrix A⁻¹ by the constants matrix B. Matrix multiplication involves summing products of rows of A⁻¹ with columns of B, resulting in the values of variables x, y, and z.
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