Given matrix $$A =\begin{bmatrix}1 & 2 & 5& 4\\0&-1&-2&-1\\ -1&1&3&0\\0&2&5&3\end{bmatrix}$$ Find matrix $B$ such that $A = adj(B)$.
Adjoint Matrix with given Matrix
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linear-algebra
matrices
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1I'm pretty sure that every introductory linear algebra book contains a chapter about adjoint of matrices. You're just required to follow the algorithm given there, nothing else. – 2017-01-29
1 Answers
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Notice that the adjugate matrix of matrix $B: 4 \times 4$ is defined such that $$B A = \left( \frac{1}{\det B} \right)I_4.$$
Now, $A = \text{adj}(B)$ is given by $$A =\begin{pmatrix}1 & 2 & 5& 4\\0&-1&-2&-1\\ -1&1&3&0\\0&2&5&3\end{pmatrix}, $$ with $\det (A) = -1$ hence it is invertible. This gives $$B = \left( \frac{1}{\det B} \right) A^{-1}.$$ Since $\det (A^{-1}) = \frac{1}{\det A} = -1,$ choosing $B = -A^{-1}$ works, because $$BA = (-A^{-1}A) = \frac{1}{\det (-A^{-1})} I_4.$$