Shortcut Methods and Tricks to Solve Numerical Problems

1. Finding the Determinant of a Matrix:

• For a 2x2 matrix, use the formula $$det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$$.
• For a 3x3 matrix, use the Sarrus rule or the Laplace expansion along any row or column.

2. Finding the Adjoint of a Matrix:

• The adjoint of a matrix is the transpose of its cofactor matrix.
• For a 2x2 matrix, the adjoint is given by $$A^{adj} = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$ where A = $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$.

3. Finding the Inverse of a Matrix:

• The inverse of a matrix is the matrix that gives you the identity matrix when multiplied by the original matrix.
• For a 2x2 matrix, the inverse is given by $$A^{-1} = \frac{1}{detA}A^{adj}$$ where A = $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ and (detA\neq 0).

4. Solving System of Linear Equations using Matrix Inversion:

• Write the system of linear equations in the form $$Ax = b$$ where A is the coefficient matrix, x is the column vector of variables, and b is the column vector of constants.
• Find the inverse of A, if it exists.
• Multiply both sides of the equation by A(^{-1}) to get (x = A^{-1}b).

5. Finding Eigenvalues and Eigenvectors of a Matrix:

• Eigenvalues are the scalar values for which the determinant of (A - \lambda I) is zero, where A is the matrix, I is the identity matrix, and (\lambda) is the eigenvalue.
• Eigenvectors are the nonzero vectors that, when multiplied by the matrix, gives a scalar multiple of themselves.

6. Finding the Rank of a Matrix:

• The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
• It can be found by reducing the matrix to its row echelon form and counting the number of nonzero rows.

7. Finding the Nullity of a Matrix:

• The nullity of a matrix is the dimension of the null space, which is the set of all solutions to the equation (Ax = 0).
• It can be found by subtracting the rank of the matrix from the number of columns in the matrix.

8. Finding the Area of a Parallelogram using Vectors:

• The area of a parallelogram formed by two vectors (\vec{a}) and (\vec{b}) is given by $$\text{Area} = |\vec{a} \times \vec{b}|$$ where (\times) denotes the cross product.

9. Finding the Volume of a Parallelepiped using Vectors:

• The volume of a parallelepiped formed by three vectors (\vec{a}, \vec{b}), and (\vec{c}) is given by $$\text{Volume} = |\vec{a} \cdot (\vec{b} \times \vec{c})|$$ where (\cdot) denotes the dot product and (\times) denotes the cross product.