Question

Matrices \(A\) and \(B\) are given. Solve the matrix equation \(A X=B\). $$ A=\left[\begin{array}{ccc} 0 & -2 & 1 \\ 0 & 2 & 2 \\ 1 & 2 & -3 \end{array}\right], \quad B=I_{3} $$

Short answer

Matrix \(A\) is not invertible; the equation has no unique solution.

Step-by-step solution

Identify Matrix Equation

The problem asks to solve for matrix \(X\) in the equation \(A X = B\). Given matrices are \(A\) and \(B\), where \(B\) is the identity matrix \(I_3\). The equation implies finding the inverse of matrix \(A\) to solve for \(X\).

Check for Inverse Existence

A matrix can be inverted if it is square and has full rank (its determinant is non-zero). Here, matrix \(A\) is a 3x3 matrix. Calculate the determinant of \(A\) to check if it is invertible.

Calculate Determinant of A

To find the determinant of \(A\), use the formula for a 3x3 matrix:\[\text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\] with the appropriate substitutions:\[\text{det}(A) = 0(2(-3)-2(2)) - (-2)(0(-3)-2(1)) + 1(0(2)-2(2))\]\[= 0 - (-4) - 4 = 0\]Thus, \(\text{det}(A) = 0\).

Analyze Invertibility

Since the determinant of matrix \(A\) is zero, matrix \(A\) is singular and cannot be inverted. Therefore, the equation \(A X = B\) does not have a unique solution.

Conclusion on Solution

Since matrix \(A\) is not invertible and there's no unique \(X\) satisfying \(A X = B\), attempts to solve this equation are infeasible without further adjustments or information.

Key concepts

Matrix Inversion

Matrix inversion is a method used to solve matrix equations of the form \(AX = B\), where \(A\) and \(B\) are known matrices. In order to solve for \(X\), it is essential to find the inverse of matrix \(A\), denoted as \(A^{-1}\). The matrix equation can then be rewritten as \(X = A^{-1}B\). This technique closely resembles the way we use division to solve linear equations in algebra.

Not all matrices can be inverted. A matrix must:

• Be a square matrix (equal number of rows and columns).
• Have a non-zero determinant.

These conditions ensure the matrix has full rank and enables the computation of an inverse.
In contexts such as this exercise, if a matrix lacks an inverse, the equation cannot be solved through inversion, as demonstrated by matrix \(A\) where its determinant equals zero.

Determinant Calculation

Calculating the determinant of a matrix is a crucial step in finding out whether it is invertible. The determinant is a special number that can be computed from a square matrix. For a 3x3 matrix like \(A\), the determinant can be calculated using the formula:\[\text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]where \(a, b, c, d, e, f, g, h,\) and \(i\) are elements of the matrix arranged in a specific manner. The zero value of the determinant indicates that the matrix does not have an inverse, which is referred to as being **singular**.

Such calculations not only determine the ability to invert matrices but also have implications in other areas such as eigenvalues and the calculations of volume transformations represented by the matrix.

Singular Matrix

A matrix is termed singular when its determinant is zero. This indicates that the matrix does not have an inverse. This property is a significant hindrance when trying to solve matrix equations like \(AX = B\), which mandates matrix \(A\) to be invertible.

In the scenario where matrix \(A\) is singular, like in this exercise, it means there's no unique solution to the equation because the matrix cannot cover an entire span of linear transformations, thereby leading to linear dependency among its rows or columns. It can result in either no solution or infinitely many solutions. Therefore, alternative methods or additional information may be necessary to find solutions to such equations.