Questions: Question Given that matrix A is defined by [[4, 7, 3], [-1, 0, 0], [2, -3, -1]], find the inverse of A if it exists.

Question Given that matrix A is defined by [[4, 7, 3], [-1, 0, 0], [2, -3, -1]], find the inverse of A if it exists.
Transcript text: Question Given that matrix $A$ is defined by $\left[\begin{array}{ccc}4 & 7 & 3 \\ -1 & 0 & 0 \\ 2 & -3 & -1\end{array}\right]$, find the inverse of $A$ if it exists. Provide your answer below: Search
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Solution

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Solution Steps

To find the inverse of a matrix, we need to ensure that the matrix is square and has a non-zero determinant. If these conditions are met, we can use a mathematical method or a library function to compute the inverse.

Step 1: Verify Matrix Properties

Matrix \( A \) is given by: \[ A = \begin{bmatrix} 4 & 7 & 3 \\ -1 & 0 & 0 \\ 2 & -3 & -1 \end{bmatrix} \] To find the inverse, we first need to ensure that \( A \) is a square matrix and that its determinant is non-zero. Since \( A \) is a \( 3 \times 3 \) matrix, it is square.

Step 2: Calculate the Determinant

The determinant of matrix \( A \) is calculated as: \[ \text{det}(A) = 1.9999999999999967 \] Since the determinant is approximately \( 2 \), which is non-zero, the inverse of \( A \) exists.

Step 3: Compute the Inverse

The inverse of matrix \( A \) is given by: \[ A^{-1} = \begin{bmatrix} -1.665 \times 10^{-16} & -1 & -1.110 \times 10^{-16} \\ -0.5 & -5 & -1.5 \\ 1.5 & 13 & 3.5 \end{bmatrix} \]

Final Answer

The inverse of matrix \( A \) is: \[ \boxed{\begin{bmatrix} -1.665 \times 10^{-16} & -1 & -1.110 \times 10^{-16} \\ -0.5 & -5 & -1.5 \\ 1.5 & 13 & 3.5 \end{bmatrix}} \]

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