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Chapter 6: Problem 53

Explain how to solve the matrix equation \(A X=B\)

Short Answer

Expert verified
The solution to the matrix equation is \(X=A^{-1}B\), where \(A^{-1}\) is the inverse of matrix \(A\)

Step by step solution

01

Confirm Matrix A is Invertible

Examine matrix \(A\) to confirm that it is invertible. This is crucial as the inverse of a singular matrix (i.e., a matrix that is not invertible) does not exist.
02

Calculate the Inverse of Matrix A

If matrix \(A\) is invertible, calculate its inverse. The inverse of a matrix \(A\) is often denoted by \(A^{-1}\). Use a suitable method such as the Gaussian elimination, adjugate, or Laplace expansion method to calculate the inverse.
03

Apply the Inverse to Both Sides of the Equation

Multiply the inverse of matrix \(A\), that is, \(A^{-1}\), to both sides of the equation. The equation now reads as \(A^{-1}AX=A^{-1}B\)
04

Simplify the Equation

Simplify the equation. Notably, the product of a matrix and its inverse yields the identity matrix. The left-hand side of the equation, \(A^{-1}A\), simplifies to the identity matrix, denoted by \(I\). Thus, \(IX=A^{-1}B\)
05

Isolate Matrix X

Finally, multiplying the identity matrix by matrix \(X\) simply yields matrix \(X\) (since \(I\) times any matrix equals the original matrix). Thus, \(X=A^{-1}B\), which is the solution to the matrix equation.

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Most popular questions from this chapter

Evaluate each determinant. $$ \left|\begin{array}{rrrr}3 & -1 & 1 & 2 \\\\-2 & 0 & 0 & 0 \\\2 & -1 & -2 & 3 \\\1 & 4 & 2 & 3\end{array}\right| $$

In applying Cramer's rule, what does it mean if \(D=0 ?\)

Determinants are used to write an equation of a line passing through two points. An equation of the line passing through the distinct points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is given by $$ \left|\begin{array}{ccc}x & y & 1 \\\x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1\end{array}\right|=0 $$ Use this information to work. Use the determinant to write an equation of the line passing through \((-1,3)\) and \((2,4) .\) Then expand the determinant, expressing the line's equation in slope-intercept form.

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} 4 & 0 \\ -3 & 5 \\ 0 & 1 \end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\ -2 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right] $$ $$ B C+C B $$

Evaluate each determinant. $$ \left|\begin{array}{rrr}2 & -4 & 2 \\\\-1 & 0 & 5 \\\3 & 0 & 4\end{array}\right| $$
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