Finding the Inverse of a 3x3 Matrix T with Determinants and Adjugates

Math Problem Statement

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Find the inverse matrix of T where T = [[-1, 2, -3], [7, 3, 0], [4, 2, 5]]

The problem asks to find the inverse of the matrix $T$ , where:

-1 & 2 & -3 \\ 7 & 3 & 0 \\ 4 & 2 & 5 \end{pmatrix}$$ ### Steps to Find the Inverse of a 3x3 Matrix Given a matrix $$T$$, its inverse $$T^{-1}$$ can be found using the following formula: $$T^{-1} = \frac{1}{\text{det}(T)} \cdot \text{adj}(T)$$ Where: - $$\text{det}(T)$$ is the determinant of $$T$$. - $$\text{adj}(T)$$ is the adjugate (or adjoint) of $$T$$, which is the transpose of the cofactor matrix. #### Step 1: Find the Determinant of $$T$$ $$\text{det}(T) = -1 \cdot \begin{vmatrix} 3 & 0 \\ 2 & 5 \end{vmatrix} - 2 \cdot \begin{vmatrix} 7 & 0 \\ 4 & 5 \end{vmatrix} + (-3) \cdot \begin{vmatrix} 7 & 3 \\ 4 & 2 \end{vmatrix}$$ Calculating the 2x2 determinants: $$\begin{vmatrix} 3 & 0 \\ 2 & 5 \end{vmatrix} = 3(5) - 0(2) = 15$$ $$\begin{vmatrix} 7 & 0 \\ 4 & 5 \end{vmatrix} = 7(5) - 0(4) = 35$$ $$\begin{vmatrix} 7 & 3 \\ 4 & 2 \end{vmatrix} = 7(2) - 3(4) = 14 - 12 = 2$$ So, $$\text{det}(T) = -1(15) - 2(35) + (-3)(2) = -15 - 70 - 6 = -91$$ #### Step 2: Find the Adjugate of $$T$$ The cofactor matrix of $$T$$ is obtained by finding the minors and applying the sign pattern: \[ \text{Cofactor Matrix} = \begin{pmatrix} \begin{vmatrix} 3 & 0 \\ 2 & 5 \end{vmatrix} & -\begin{vmatrix} 7 & 0 \\ 4 & 5 \end{vmatrix} & \begin{vmatrix} 7 & 3 \\ 4 & 2 \end{vmatrix} \\ -\begin{vmatrix} 2 & -3 \\ 2 & 5 \end{vmatrix} & \begin{vmatrix} -1 & -3 \\ 4 & 5 \end{vmatrix} & -\begin{vm

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Linear Algebra
Matrix Inversion
Determinants

Inverse of a matrix T: T^{-1} = 1/det(T) * adj(T)
Determinant of a 3x3 matrix
Adjugate (adjoint) of a matrix

Properties of determinants
Cofactor expansion for determinants
Matrix inversion theorem

Undergraduate level

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