Question

Matrices \(A\) and \(B\) are given. Solve the matrix equation \(A X=B\). $$ \begin{aligned} A=&\left[\begin{array}{ccc} -5 & -4 & -1 \\ 8 & -2 & -3 \\ 6 & 1 & -8 \end{array}\right] \\ B=\left[\begin{array}{ccc} -21 & -8 & -19 \\ 65 & -11 & -10 \\ 75 & -51 & 33 \end{array}\right] \end{aligned} $$

Short answer

Find the inverse of \(A\), then multiply by \(B\) to get \(X\).

Step-by-step solution

Check if A is Invertible

To solve the equation \(A X = B\), we first need to check if matrix \(A\) is invertible. A matrix is invertible if its determinant is non-zero. Calculate the determinant of \(A\).

Calculate Determinant of A

Find the determinant of \(A\) using the formula \( \text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \), where \[ A = \left[ \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right] \].

Apply Determinant Formula

Substitute the elements of \(A\) into the determinant formula: \[ \text{det}(A) = (-5)(-2 \cdot -8 - -3 \cdot 1) - (-4)(8 \cdot -8 - -3 \cdot 6) + (-1)(8 \cdot 1 - -2 \cdot 6) \]. Compute the value.

Evaluate Determinant

Calculate each term: \((-5)(16 + 3)\), \(- (-4)(-64 - 18)\), \(- (-1)(8 + 12)\). Simplify to find \( \text{det}(A) = -95 + 328 - 20 = 213 \). Since the determinant is non-zero (213), \(A\) is invertible.

Calculate Inverse of A

Since \(A\) is invertible, find its inverse \(A^{-1}\) using the formula for the inverse of a 3x3 matrix. This involves dividing the adjugate of \(A\) by \( \text{det}(A) \).

Multiply A Inverse with B

Once \(A^{-1}\) is found, solve for \(X\) by multiplying \(A^{-1}\) with \(B\): \(X = A^{-1}B\). Perform the matrix multiplication to get the result matrix \(X\).

Check Solution

Ensure that the product \(AX\) equals \(B\). Re-multiply \(A\) by \(X\) to confirm \(AX = B\).

Key concepts

Matrix Inversion

Matrix inversion is the process of finding a matrix that, when multiplied by the original matrix, results in the identity matrix. It is a crucial concept when solving matrix equations like \(A X = B\). In the context of this equation, if matrix \(A\) has an inverse, denoted as \(A^{-1}\), then we can solve for \(X\) by multiplying \(B\) by \(A^{-1}\).

To find the inverse of a 3x3 matrix, follow these steps:

• Calculate the determinant of the matrix.
• If the determinant is non-zero, proceed to find the adjugate of the matrix.
• Divide the adjugate by the determinant.

When dealing with larger matrices, finding an inverse can be more complex, but these basic principles still apply.

Ensure that operations are carefully verified as precision matters; a small mistake can lead to a completely different outcome.

Determinant Calculation

The determinant is a special number calculated from a square matrix that provides significant insights into the matrix's properties.
For a 3x3 matrix like \(A = \left[ \begin{array}{ccc} -5 & -4 & -1 \ 8 & -2 & -3 \ 6 & 1 & -8 \end{array} \right]\), the determinant is found using the formula:\[\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]Here:

• \(a, b, c\) are the elements of the first row.
• \(d, e, f\) from the second row.
• \(g, h, i\) from the third row.

In the given example, substituting the respective values from matrix \(A\), one computes the determinant, which tells whether the matrix is invertible (determinant is non-zero). Calculating the determinant correctly ensures that the solution process is valid, as it directly affects whether we can proceed to find the matrix inverse.

Solving Linear Systems

When solving linear systems represented by matrix equations like \(AX = B\), it is important to comprehend the role of matrix inversion and determinants in this process.
For the equation to be solved, matrix \(A\) must be invertible. The absence of an inverse means no unique solution can be found using this technique.

Once ensured that matrix \(A\) is invertible (determinant not zero), the steps to solve for \(X\) include:

• Calculating \(A^{-1}\), the inverse of \(A\).
• Performing matrix multiplication of \(A^{-1}\) with \(B\).

This process results in the matrix \(X\), which is the solution to our system.
Verifying the solution by re-multiplying \(A\) and \(X\) is crucial to confirm \(AX = B\). If there is an error or adjustment needed, it can be traced back to any initial miscalculation in steps one or two. Understanding each part thoroughly helps avoid common mistakes and leads to precise and accurate solutions.