Minors, Cofactors & Adjoint

MATMinors, Cofactors & Adjoint

A minor $M_{ij}$ is the determinant of the submatrix left after deleting row $i$ and column $j$ , and the cofactor $C_{ij} = (-1)^{i+j}M_{ij}$ attaches the chequerboard sign. Assembling the cofactors into a matrix and transposing gives the adjoint, the bridge to determinant expansion and to the inverse.

ISC asks students to list specific minors and cofactors or to verify $A(\text{adj}\,A) = |A|I$ , so accuracy with signs and the final transpose is essential.

Minor and cofactor

C_{ij} = (-1)^{i+j} M_{ij}

M_{ij}

is the minor (determinant after deleting row

i

, column

j

);

C_{ij}

is its signed cofactor.

Sign chequerboard

\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}

The pattern of

(-1)^{i+j}

for a

3\times3

matrix; '+' keeps the minor, '-' negates it.

Adjoint as transpose of cofactors

\text{adj}\,A = [C_{ij}]^{T} = \begin{pmatrix} C_{11} & C_{21} & C_{31} \\ C_{12} & C_{22} & C_{32} \\ C_{13} & C_{23} & C_{33} \end{pmatrix}

Each entry of

\text{adj}\,A

in position

(i,j)

is the cofactor

C_{ji}

(note the swapped indices).

Adjoint verification property

A\,(\text{adj}\,A) = (\text{adj}\,A)\,A = |A|\,I

I

is the identity matrix; this is the standard property students are asked to verify.

To get $M_{ij}$ , delete the $i$ -th row and $j$ -th column and evaluate the determinant of what remains.
Convert each minor to a cofactor by multiplying with $(-1)^{i+j}$ ; even index-sums keep the sign, odd ones flip it.
The cofactor matrix has $C_{ij}$ in position $(i,j)$ ; the adjoint is its transpose, so $\text{adj}\,A$ has $C_{ji}$ in position $(i,j)$ .
For a $2\times2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , $\text{adj}\,A = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ .
Useful results: $|\text{adj}\,A| = |A|^{n-1}$ and $\text{adj}(AB) = (\text{adj}\,B)(\text{adj}\,A)$ for $n\times n$ matrices.
Verifying $A(\text{adj}\,A) = |A|I$ means every diagonal entry of the product equals $|A|$ and every off-diagonal entry is $0$ .
The expansion $|A| = a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3}$ uses cofactors of the same row; pairing a row's elements with another row's cofactors gives $0$ .

Where the marks go

Confusing the minor with the cofactor by forgetting the $(-1)^{i+j}$ sign factor.
Forgetting to transpose: handing in the cofactor matrix instead of its transpose as $\text{adj}\,A$ .
Multiplying an element of one row by the cofactor of a different row and expecting $|A|$ , when this 'alien cofactor' sum is actually $0$ .
Sign-pattern errors at the off-diagonal positions, where $(-1)^{i+j}$ is negative.

How the board asks it

Numericalminor and cofactor definition
For the matrix $A=\begin{pmatrix} 2 & -1 & 3 \\ 0 & 4 & -2 \\ 1 & 5 & 6 \end{pmatrix}$ , find the minors $M_{21}$ and $M_{32}$ and the cofactors $C_{21}$ and $C_{32}$ .
Numericaladjoint as transpose of the cofactor matrix
Find the adjoint of the matrix $A=\begin{pmatrix} 1 & 2 & -1 \\ 3 & 0 & 2 \\ -1 & 1 & 4 \end{pmatrix}$ .
Derive / prove $A(\text{adj}\,A) = |A|\,I$
For $A=\begin{pmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{pmatrix}$ , verify that $A(\text{adj}\,A) = (\text{adj}\,A)A = |A|\,I$ .
Numerical $|\text{adj}\,A| = |A|^{n-1}$
If $A$ is a $3 \times 3$ matrix with $|A| = 4$ , find the value of $|\text{adj}\,A|$ .
Give reasonssum of products of elements with cofactors of another row is zero
Without expanding the determinant, explain why $a_{11}C_{21} + a_{12}C_{22} + a_{13}C_{23} = 0$ for a $3 \times 3$ matrix $A$ , where $C_{2j}$ denotes the cofactor of the entry $a_{2j}$ .
Multiple choice $2\times2$ adjoint formula
If $A=\begin{pmatrix} 3 & -2 \\ 4 & 1 \end{pmatrix}$ , then $\text{adj}\,A$ equals: (a) $\begin{pmatrix} 1 & 2 \\ -4 & 3 \end{pmatrix}$ , (b) $\begin{pmatrix} 1 & -2 \\ 4 & 3 \end{pmatrix}$ , (c) $\begin{pmatrix} 3 & 4 \\ -2 & 1 \end{pmatrix}$ , (d) $\begin{pmatrix} -1 & 2 \\ -4 & -3 \end{pmatrix}$ .

Practise this topic

PreviousEvaluating Determinants & Using Properties to Prove Results NextInverse of a Matrix using Adjoint

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.