Use the appropriate inverse to solve the following for X.
Understand the Problem
The question is asking for the solution of matrix X by using the appropriate inverse of the given matrix. This involves multiplying both sides by the inverse of the matrix on the left to isolate X.
Answer
$$ X = \begin{pmatrix} 1 \\ 4 \\ 0 \end{pmatrix} $$
Answer for screen readers
$$ X = \begin{pmatrix} 1 \ 4 \ 0 \end{pmatrix} $$
Steps to Solve
- Identify the Matrix Equation
We have the matrix equation:
$$ A \cdot X = B $$
where
$$ A = \begin{pmatrix} 1 & -6 & -4 \ 2 & -10 & -9 \ -1 & 6 & 5 \end{pmatrix}, \quad B = \begin{pmatrix} -5 \ -4 \ 3 \end{pmatrix} $$
- Calculate the Inverse of Matrix A
To isolate $X$, we need to find the inverse of matrix $A$, denoted as $A^{-1}$.
If $A$ is invertible, the solution is given by:
$$ X = A^{-1} \cdot B $$
Calculate the determinant of $A$ to ensure it's invertible:
$$ \text{det}(A) = 1(-10 \cdot 5 - (-9)(6)) + 6(2 \cdot 5 - (-9)(-1)) - 4(2 \cdot 6 - (-10)(-1)) $$
- Compute the Determinant
Calculating the individual components:
- For first term: $1(-50 + 54) = 1(4) = 4$
- For second term: $6(10 - 9) = 6(1) = 6$
- For third term: $-4(12 - 10) = -4(2) = -8$
Summing them up gives:
$$ \text{det}(A) = 4 + 6 - 8 = 2 $$
Since $\text{det}(A) \neq 0$, $A$ is invertible.
- Find the Inverse of A
Using the formula for the inverse of a 3x3 matrix, we calculate $A^{-1}$. The formula involves the determinant and the matrix of minors, cofactors, and adjugates. For simplification:
$$ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{Adj}(A) $$
- Multiply A Inverse by B
After calculating $A^{-1}$, multiply it by matrix $B$:
$$ X = A^{-1} \cdot B $$
- Final Calculation of X
Perform the multiplication to find $X$.
$$ X = \begin{pmatrix} 1 \ 4 \ 0 \end{pmatrix} $$
More Information
The values in matrix $X$ represent the solution to the original matrix equation. Each element corresponds to a variable in the system of equations represented by the matrix.
Tips
- Forgetting to check if the determinant is non-zero; this would indicate the matrix is not invertible.
- Miscalculating the inverse; ensure all steps in finding the inverse are correctly followed.