Determinants — Part 2: Adjoint, inverse and systems of equations: visual lesson, CBSE Class 12 Mathematics

Lesson

Determinants, Part 2: inverses and systems

From minors to the solution of a system, one step at a time.

In this lesson you will

Part 2 in one line: from minors to the solution of a system.

Concept

Concept

The minor $M_{ij}$ is the determinant left after deleting row $i$ and column $j$ . The cofactor is $A_{ij} = (-1)^{i + j}M_{ij}$ .

Minor: delete and evaluate. Cofactor: minor with the chessboard sign.
Term	How to get it	Example in $\begin{vmatrix} 2 & 1 & 3 \\ 1 & -1 & 0 \\ 0 & 2 & 1 \end{vmatrix}$
Minor $M_{21}$	delete row 2 and column 1, evaluate	$\begin{vmatrix} 1 & 3 \\ 2 & 1 \end{vmatrix} = -5$
Cofactor $A_{21}$	$(-1)^{2 + 1}M_{21}$	$5$
Minor $M_{33}$	delete row 3 and column 3	$\begin{vmatrix} 2 & 1 \\ 1 & -1 \end{vmatrix} = -3$
Cofactor $A_{33}$	$(-1)^{6}M_{33}$	$-3$