Lesson 6 of 11 · 7 min
The adjoint
NCERT §4.5.1
Arrange all the cofactors in a grid, flip it over its diagonal, and something remarkable happens when it multiplies the original matrix.
The lesson in notes
In short
The adjoint of a square matrix A, written adj A, is the transpose of its matrix of cofactors: the (i, j) entry of adj A is Aⱼᵢ.
So the cofactors of row 1 of A go down column 1 of adj A, the cofactors of row 2 down column 2, and so on.
Worked example: A = [[2, 3], [1, 4]] has cofactors A₁₁ = 4, A₁₂ = −1, A₂₁ = −3, A₂₂ = 2, so adj A = [[4, −3], [−1, 2]].
Shortcut for order 2: adj [[a, b], [c, d]] = [[d, −b], [−c, a]]. Swap the two diagonal entries and change the sign of the other two.
Key theorem (stated without proof in NCERT): A(adj A) = (adj A)A = |A| I for every square matrix A of order n.
Why it holds: entry (i, k) of A(adj A) pairs row i of A with the cofactors of row k. For i = k that is the expansion of |A|; for i ≠ k it is a row paired with another row's cofactors, which is 0.