TN 12th Maths — Learn, Visualize, Practice

1. Minors, cofactors and the adjoint

To invert a matrix we first need two building blocks. The minor $M_{ij}$ of an entry is the determinant of the smaller matrix left after deleting row $i$ and column $j$ . The cofactor attaches a checkerboard sign to that minor.

DefinitionCofactor and adjoint

The cofactor of entry $a_{ij}$ is $C_{ij} = (-1)^{i+j} M_{ij}$ . The adjoint (adjugate) of $A$ is the transpose of the matrix of cofactors: $\operatorname{adj}(A) = [C_{ij}]^{T}$ .

Note: The transpose matters: the $(i,j)$ entry of $\operatorname{adj}(A)$ is $C_{ji}$, not $C_{ij}$. For a $2\times2$ matrix this simplifies to swapping the diagonal and negating the off-diagonal: $\operatorname{adj}\!\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.

2. Inverse of a non-singular matrix

A square matrix is called non-singular when its determinant is non-zero. Exactly these matrices can be inverted, and the adjoint gives an explicit formula.

DefinitionInverse via adjoint

If $|A| \ne 0$ , then $A$ is invertible and $A^{-1} = \dfrac{1}{|A|}\,\operatorname{adj}(A)$ , satisfying $AA^{-1} = A^{-1}A = I$ .

Key idea: If $|A| = 0$ the matrix is singular and has no inverse — you would be dividing by zero. Try it live in the Visualize tab by making two rows proportional.

Example

Find the inverse of

A = \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}

.

3. Determinant & adjoint identities

These identities appear constantly in objective questions — knowing them saves you from computing full matrices.

$A\,(\operatorname{adj} A) = (\operatorname{adj} A)\,A = |A|\,I$
$|\operatorname{adj}(A)| = |A|^{\,n-1}$ for an $n\times n$ matrix
$|A^{-1}| = \dfrac{1}{|A|}$
$(AB)^{-1} = B^{-1}A^{-1}$ (order reverses)
$\operatorname{adj}(AB) = \operatorname{adj}(B)\,\operatorname{adj}(A)$

Example

If

A

is

3\times3

with

|A| = 2

, find

|\operatorname{adj}(A)|

.

4. Elementary transformations & Gauss–Jordan

Row operations — swapping two rows, scaling a row, or adding a multiple of one row to another — do not change the solution set of a system. Applying them to $[A \mid I]$ until the left block becomes $I$ turns the right block into $A^{-1}$ . This is the Gauss–Jordan method, and it is often faster than the adjoint for $3\times3$ matrices.

[\,A \mid I\,] \;\xrightarrow{\text{row operations}}\; [\,I \mid A^{-1}\,]

5. Solving systems: the matrix-inversion method

A linear system can be packaged as a single matrix equation. If the coefficient matrix is non-singular, one multiplication solves the whole system at once.

System in matrix form

AX = B \quad\Longrightarrow\quad X = A^{-1}B

Example

Solve

2x + 5y = 1,\ x + 3y = 2

using

X = A^{-1}B

.

6. Cramer's rule

When you only need the values (not $A^{-1}$ itself), Cramer's rule expresses each unknown directly as a ratio of determinants.

x_i = \dfrac{|A_i|}{|A|}, \quad |A| \ne 0

where $A_i$ is $A$ with its $i$ -th column replaced by the column $B$ .

7. Consistency of a system by rank

Not every system has a solution. The rank of a matrix — the number of non-zero rows after reducing to echelon form — decides everything. Compare the rank of $A$ with that of the augmented matrix $[A \mid B]$ .

$\rho(A) = \rho([A\mid B]) = n$ → a unique solution
$\rho(A) = \rho([A\mid B]) < n$ → infinitely many solutions
$\rho(A) \ne \rho([A\mid B])$ → no solution (inconsistent)

Watch out: A homogeneous system $AX = 0$ is always consistent (it has the trivial solution $X = 0$). It has non-trivial solutions precisely when $\rho(A) < n$, i.e. when $|A| = 0$.