Compute the inverse of the diagonal matrix [2 0 0; 0 -4 0; 0 0 5].

[(1)/(2) 0 0; 0 - (1)/(4) 0; 0 0 (1)/(5)]

[(1)/(2) 0 0; 0 (1)/(4) 0; 0 0 (1)/(5)]

Exercise · ZW-INVERSE-MATRIX-003 · ★☆☆ Basic

Compute the inverse of the diagonal matrix \(\begin{pmatrix}2 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 5\end{pmatrix}\).

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C. \(\begin{pmatrix}\frac{1}{2} & 0 & 0 \\ 0 & - \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{5}\end{pmatrix}\)

Explanation

The inverse of a diagonal matrix with nonzero entries is the diagonal of reciprocals: \(\frac12,\,-\frac14,\,\frac15\). The sign is kept because the reciprocal of a negative number is negative.

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Review the concept: Inverse matrix

The inverse of a square matrix \(A\) is the matrix \(A^{-1}\) that gives the identity \(I\) when multiplied by \(A\) on either side; if it exists, \(A\) is called invertible.

\[AA^{-1}=A^{-1}A=I\]

Compute the inverse of the diagonal matrix \(\begin{pmatrix}2 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 5\end{pmatrix}\).

Explanation

Review the concept: Inverse matrix

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