### Shortcut Methods

**JEE Mains:**

**1.** Finding the inverse of a matrix:

**Shortcut:**

- Use the formula for 2x2 matrix in finding inverse of matrix A. $$A^{-1} =\frac{1}{|A|}A^{adj}$$

**2.** Determining the rank of a matrix:

**Shortcut:**

- Echelon form of the matrix can be directly used in finding the rank of a matrix. The count of non zero rows in echelon form is the required rank.

**3.** Solving a system of linear equations using matrices:
**Shortcut:**

- To solve a system of linear equations using matrices, you can use the formula $$X= A^{-1}B$$ Where X is column matrix of variables to be determined i.e. $$ X=\begin{bmatrix} x \\ y \end{bmatrix} $$ matrix A is the coefficient matrix which is of order (n x n ) i.e $$ A=\begin{bmatrix} a & b \\ c & d \end{bmatrix} $$ matrix B is column matrix of constants i.e $$B=\begin{bmatrix} e \\ f \end{bmatrix}$$

**4.** Finding the eigenvalues and eigenvectors of a matrix:

- Use the characteristic equation and determinants in order to find the Eigen values and corresponding Eigen vectors.

**5.** Calculating the determinant of a matrix:
**Shortcut:**

- Sarrus Rule.
- Formula for 3x3 determinant (expansion along the first row).

**CBSE Boards:**

**1.** Solving linear equations using matrices:

- Use the formula : $$X= A^{-1}B$$ Where X is column matrix of variables to be determined i.e. $$ X=\begin{bmatrix} x \\ y \end{bmatrix} $$ matrix A is the coefficient matrix which is of order (n x n ) i.e $$ A=\begin{bmatrix} a & b \\ c & d \end{bmatrix} $$ matrix B is column matrix of constants i.e $$B=\begin{bmatrix} e \\ f \end{bmatrix}$$

**2.** Finding the multiplicative inverse of a matrix:

- Use the formula for 2x2 matrix in finding multiplicative inverse. $$A^{-1} =\frac{1}{|A|}A^{adj}$$

**3.** Transposing a matrix:
**Shortcut:**

- For Transpose of a matrix. Just take the mirror image of the matrix about the principal diagonal.

**4.** Calculating the determinant of a square matrix:
**Shortcut:**

- For 3x3 matrices , determinant can be expanded along C1.
- Matrix of order nxn determinant can be expanded along any row or column.

**5.** Finding the adjoint of a matrix:
**Shortcut:**

- For 2x2 matrices Adjoint is equal to the transpose of cofactors matrix.