Question 1.9.2: Compute the determinant of the matrix

Differential Equations with Linear Algebra [EXP-50954]

Compute the determinant of the matrix

$A=\begin{bmatrix}2& -2 &7 \\ 0 & -3 &5 \\ 0&0&4 \end{bmatrix}$

In addition, determine if A is invertible.

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Again using the definition, we see that

$det A=2det \begin{bmatrix} -5 & 3 \\ 0 & 4\end{bmatrix} −(−2) det \begin{bmatrix} 0 & 3 \\ 0 & 4\end{bmatrix} +7det \begin{bmatrix} 0 & -5 \\ 0 & 0\end{bmatrix}$

= 2(−5 · 4−2 · 0)+2(0−0)+7(0−0)
= 2(−5)(4)=−40

Note particularly that the determinant of A is the product of its diagonal entries.Moreover, A clearly has a pivot position in every row, and so by this fact (or equivalently by the nonzero determinant of A) we see that A is invertible.

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