Define determinant. Compute the determinant without expanding \(\begin{bmatrix}-2 & 8 & -9\\ -1 & 7 & 0\\ 1 & -4 & 2\end{bmatrix}\)

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The determinant of a matrix is a scalar value that is a function of the entries of a square matrix. It is basically a special number that is calculated from a square matrix.

Given

\(\begin{vmatrix}-2 & 8 & -9\\ -1 & 7 & 0\\ 1 & -4 & 2\end{vmatrix}\)

Apply, R1 ↔ R3

\(~ \begin{vmatrix} 1 & -4 & 2\\ -1 & 7 & 0\\-2 & 8 & -9\end{vmatrix}\)

Apply R2 → R2 + R1,   R3 → R3 + 2R1

\(~ \begin{vmatrix} 1 & -4 & 2\\ 0 & 3 & 2\\ 0 & 0 & -5\end{vmatrix}\)

Since, it is an upper triangular matrix

Determinant  = -(1 x 3 x -5) = 15

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