Adjoint identity
\[A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|\,I\]
Consequences: \(A^{-1} = \dfrac{\operatorname{adj}A}{|A|}\) and \(|\operatorname{adj}A| = |A|^{n-1}\).
Determinants
Adjoint and Inverse of a Matrix
Understand
The adjoint of a square matrix is the transpose of its cofactor matrix: \(\operatorname{adj}A = [A_{ij}]^T\).
The fundamental identity is
\[A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|\,I.\]
So if \(|A| \ne 0\), \(A\) is invertible and
\[A^{-1} = \frac{1}{|A|}\operatorname{adj}A.\]
If \(|A| = 0\), \(A\) has no inverse.
Useful results (for \(n\times n\) matrices)
- \(|\operatorname{adj}A| = |A|^{n-1}\)
- \(|A^{-1}| = \dfrac{1}{|A|}\)
- 2×2 shortcut: \(\operatorname{adj}\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\)
Key Concepts
- Compute \(|A|\) first; if 0, stop – no inverse.
- Do not forget to transpose the cofactor matrix.
Formula Bank
Inverse using adjoint
\[A^{-1} = \frac{1}{|A|}\operatorname{adj}A, \quad |A| \ne 0\]
Also \(|A^{-1}| = 1/|A|\).
Adjoint identities
\[A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|I,\qquad |\operatorname{adj}A|=|A|^{n-1}\]
Key Points
Check |A| first
Before finding an inverse, evaluate |A|. If it is zero, the inverse does not exist – say so and stop.
Common Mistakes
Forgetting to transpose the cofactor matrix
Taking the cofactor matrix itself as adj A.
Correct: \(\operatorname{adj}A = (\text{cofactor matrix})^T\). The (1,2) entry of adj A is the cofactor \(A_{21}\).
Practice & Topic Test
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\(\operatorname{adj}A\) = transpose of cofactor matrix. \(A\,\operatorname{adj}A = |A|I\). \(A^{-1} = \operatorname{adj}A / |A|\) when \(|A| \ne 0\). \(|\operatorname{adj}A| = |A|^{n-1}\).
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