Questions: Let A = [-2 -7 -9; 2 5 6; 1 3 4]. Find the third column of A^-1 without computing the other columns.

Transcript text: 45. Let $A=\left[\begin{array}{rrr}-2 & -7 & -9 \\ 2 & 5 & 6 \\ 1 & 3 & 4\end{array}\right]$. Find the third column of $A^{-1}$ without computing the other columns.

Solution

Solution Steps

To find the third column of the inverse of matrix $ A $, we can use the property that the columns of $ A^{-1} $ are the solutions to the system of equations $ A \mathbf{x} = \mathbf{e}_i $, where $ \mathbf{e}_i $ is the standard basis vector. For the third column, we solve $ A \mathbf{x} = \mathbf{e}_3 $.

Step 1: Define the Matrix and Standard Basis Vector

We start with the matrix $ A $ defined as: \[ A = \begin{bmatrix} -2 & -7 & -9 \\ 2 & 5 & 6 \\ 1 & 3 & 4 \end{bmatrix} \] We also define the standard basis vector for the third column: \[ \mathbf{e}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \]

Step 2: Solve the System of Equations

To find the third column of the inverse of $ A $, we need to solve the equation: \[ A \mathbf{x} = \mathbf{e}_3 \] This results in the following system of equations: \[ \begin{bmatrix} -2 & -7 & -9 \\ 2 & 5 & 6 \\ 1 & 3 & 4 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}

\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \]

Step 3: Find the Solution

Upon solving the system, we find: \[ \mathbf{x} = \begin{bmatrix} 3 \\ -6 \\ 4 \end{bmatrix} \] Thus, the third column of $ A^{-1} $ is: \[ \begin{bmatrix} 3 \\ -6 \\ 4 \end{bmatrix} \]