Math Problem Statement
Compute the determinant of the 4x4 matrix: [[a, 1, 0, 0], [-1, b, 1, 0], [0, -1, c, 1], [0, 0, -1, d]]
Solution
You have provided a matrix, and the task is to compute its determinant. The matrix is a 4x4 matrix given by:
a & 1 & 0 & 0 \\ -1 & b & 1 & 0 \\ 0 & -1 & c & 1 \\ 0 & 0 & -1 & d \end{pmatrix}$$ To compute the determinant of this 4x4 matrix, we will use cofactor expansion along the first row, which contains two zeros and simplifies the computation: $$\det(A) = a \cdot \det \begin{pmatrix} b & 1 & 0 \\ -1 & c & 1 \\ 0 & -1 & d \end{pmatrix} - 1 \cdot \det \begin{pmatrix} -1 & 1 & 0 \\ 0 & c & 1 \\ 0 & -1 & d \end{pmatrix}$$ ### Step 1: Compute the determinant of the 3x3 matrix We first compute the determinant of the matrix: $$\det \begin{pmatrix} b & 1 & 0 \\ -1 & c & 1 \\ 0 & -1 & d \end{pmatrix}$$ This determinant can be computed using the rule for a 3x3 matrix: $$\det = b \cdot \det \begin{pmatrix} c & 1 \\ -1 & d \end{pmatrix} - 1 \cdot \det \begin{pmatrix} -1 & 1 \\ 0 & d \end{pmatrix}$$ $$= b \cdot (cd - (-1)) - (d \cdot (-1) - 1 \cdot 0) = b \cdot (cd + 1) - (-d) = b(cd + 1) + d$$ So the determinant of the 3x3 matrix is: $$b(cd + 1) + d$$ ### Step 2: Compute the determinant of the second 3x3 matrix Now, compute the determinant of: $$\det \begin{pmatrix} -1 & 1 & 0 \\ 0 & c & 1 \\ 0 & -1 & d \end{pmatrix}$$ Using the cofactor expansion along the first row: $$= -1 \cdot \det \begin{pmatrix} c & 1 \\ -1 & d \end{pmatrix} = -1 \cdot (cd + 1) = -(cd + 1)$$ ### Step 3: Final determinant calculation Now we substitute the values back into the original cofactor expansion: \[ \det(AAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Cofactor Expansion
Formulas
det(A) = a * det(3x3 matrix) - 1 * det(3x3 matrix)
Determinant of 3x3 matrix = b(cd + 1) + d
Determinant of another 3x3 matrix = -(cd + 1)
Theorems
Cofactor expansion
Suitable Grade Level
Undergraduate