Matrices And Determinants

Introduction

Matrices and determinants are fundamental concepts in linear algebra, which is a crucial part of the JEE Main Mathematics syllabus. These topics have wide applications in various fields such as physics, computer science, and engineering. In this study note, we'll break down these concepts into digestible parts, providing detailed explanations, examples, and tips to help you master them.

Matrices

Definition and Types of Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. The numbers in the matrix are called elements or entries.

Types of Matrices

1. Row Matrix: A matrix with only one row.
2. Column Matrix: A matrix with only one column.
3. Square Matrix: A matrix with the same number of rows and columns.
4. Diagonal Matrix: A square matrix where all elements outside the main diagonal are zero.
5. Scalar Matrix: A diagonal matrix where all the diagonal elements are the same.
6. Identity Matrix: A diagonal matrix where all the diagonal elements are 1.
7. Zero Matrix: A matrix where all elements are zero.

Example

Consider the following matrices:

• Row Matrix: $A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$

• Column Matrix: $B = \begin{bmatrix} 4 \ 5 \ 6 \end{bmatrix}$

• Square Matrix: $C = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$

• Diagonal Matrix: $D = \begin{bmatrix} 1 & 0 \ 0 & 2 \end{bmatrix}$

• Identity Matrix: $I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$

• Zero Matrix: $O = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$

Matrix Operations

Addition and Subtraction

Matrices can be added or subtracted if they have the same dimensions. The operations are performed element-wise.

$$ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} $$

$$ A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix} $$

Scalar Multiplication

Each element of the matrix is multiplied by a scalar value.

$$ kA = k \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} ka_{11} & ka_{12} \ ka_{21} & ka_{22} \end{bmatrix} $$

Matrix Multiplication

The product of two matrices $A$ (of size $m \times n$) and $B$ (of size $n \times p$) is a matrix $C$ (of size $m \times p$).

$$ C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} $$

Example

Let $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 0 \ 1 & 3 \end{bmatrix}$. Then,

$$ AB = \begin{bmatrix} (1 \cdot 2 + 2 \cdot 1) & (1 \cdot 0 + 2 \cdot 3) \ (3 \cdot 2 + 4 \cdot 1) & (3 \cdot 0 + 4 \cdot 3) \end{bmatrix} = \begin{bmatrix} 4 & 6 \ 10 & 12 \end{bmatrix} $$

Common Mistake

Matrix multiplication is not commutative, i.e., $AB \neq BA$ in general.

Transpose of a Matrix

The transpose of a matrix $A$ is obtained by swapping its rows with columns.

$$ A^T = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}^T = \begin{bmatrix} a_{11} & a_{21} \ a_{12} & a_{22} \end{bmatrix} $$

Inverse of a Matrix

A square matrix $A$ has an inverse $A^{-1}$ if and only if $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix.

Finding the Inverse

For a $2 \times 2$ matrix:

$$ A = \begin{bmatrix} a & b \ c & d \end{bmatrix}, \quad A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} $$

Note

The determinant $ad - bc$ must be non-zero for the inverse to exist.

Determinants

Definition

The determinant is a scalar value that is a function of the entries of a square matrix. It provides important properties of the matrix, such as whether it is invertible.

Determinant of a $2 \times 2$ Matrix

For a matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, the determinant is given by:

$$ \text{det}(A) = ad - bc $$

Determinant of a $3 \times 3$ Matrix

For a matrix $A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$, the determinant is given by:

$$ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) $$

Properties of Determinants

1. Multiplicative Property: $\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)$
2. Transpose Property: $\text{det}(A^T) = \text{det}(A)$
3. Row Operations:
   • Swapping two rows changes the sign of the determinant.
   • Multiplying a row by a scalar multiplies the determinant by that scalar.
   • Adding a multiple of one row to another row does not change the determinant.

Applications of Determinants

Solving Systems of Linear Equations

Using Cramer's Rule, a system of linear equations can be solved if the determinant of the coefficient matrix is non-zero.

Example

Consider the system of equations: $$ \begin{cases} ax + by = e \ cx + dy = f \end{cases} $$

The solution is given by: $$ x = \frac{\text{det}(A_x)}{\text{det}(A)}, \quad y = \frac{\text{det}(A_y)}{\text{det}(A)} $$

where $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, $A_x = \begin{bmatrix} e & b \ f & d \end{bmatrix}$, $A_y = \begin{bmatrix} a & e \ c & f \end{bmatrix}$.

Conclusion

Matrices and determinants are powerful tools in mathematics, with wide-ranging applications in various fields. Mastery of these topics is essential for success in JEE Main Mathematics. Practice various types of problems, understand the properties, and apply the concepts to solve complex problems efficiently.

Tip

Regular practice and solving previous years' JEE Main questions can significantly boost your understanding and performance in these topics.