Math Problem Statement

M 1 k 0 k 1 0 0 0 1

M^{-1}

Solution

Let us correctly compute M−1M^{-1} for the matrix:

\begin{bmatrix} 1 & k & 0 \\ k & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$ --- ### Step 1: Determinant of $$M$$ The determinant of $$M$$ is: $$\det(M) = \begin{vmatrix} 1 & k & 0 \\ k & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}.$$ Expanding along the third row: $$\det(M) = 1 \cdot \begin{vmatrix} 1 & k \\ k & 1 \end{vmatrix}.$$ Compute the determinant of the $$2 \times 2$$ matrix: \[ \det\begin{bmatrix}

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inversion
Determinants

Formulas

Determinant formula: det(M) = |A|
Matrix inverse formula: M^{-1} = adj(M) / det(M)

Theorems

Properties of determinants
Formula for inverse of a 3x3 matrix

Suitable Grade Level

Grades 10-12 and College

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