Matrices and Determinants Formulas For JEE 2027, Check PDF

Matrix Notation

$$$A = [a_{ij}]_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$$

• $$a_{ij}$$ = element at row $$i$$, column $$j$$
• The order of the matrix is $$m \times n$$

Special Types of Matrices

Triangular Matrices

• Upper triangular: All elements below the main diagonal are zero ($$a_{ij} = 0$$ for $$i > j$$).
• Lower triangular: All elements above the main diagonal are zero ($$a_{ij} = 0$$ for $$i < j$$).

If $$A = [a_{ij}]$$ and $$B = [b_{ij}]$$ are both $$m \times n$$ matrices:

$$$A + B = [a_{ij} + b_{ij}]_{m \times n}$$$

If $$k$$ is a scalar: $$kA = [k \cdot a_{ij}]$$ (every element is multiplied by $$k$$).

If $$A$$ is $$m \times p$$ and $$B$$ is $$p \times n$$, then $$AB$$ is $$m \times n$$ with:

$$$(AB)_{ij} = \sum_{k=1}^{p} a_{ik} \cdot b_{kj}$$$

• $$AB$$ exists only if columns of $$A$$ = rows of $$B$$
• In general, $$AB \neq BA$$ (matrix multiplication is not commutative)
• Matrix multiplication is associative: $$(AB)C = A(BC)$$
• $$AI = IA = A$$ (identity is the multiplicative identity)

Properties of Transpose

• $$(A^T)^T = A$$
• $$(A + B)^T = A^T + B^T$$
• $$(kA)^T = kA^T$$
• $$(AB)^T = B^T A^T$$ (note the reversal of order!)

Key Results

• Every square matrix can be written as the sum of a symmetric and a skew-symmetric matrix: $$$A = \frac{A + A^T}{2} + \frac{A - A^T}{2}$$$
• For a skew-symmetric matrix of odd order, $$\det(A) = 0$$.

$$2 \times 2$$ Determinant

$$$\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$$$

$$3 \times 3$$ Determinant (Expansion along Row 1)

$$$\det\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2)$$$

Properties of Determinants

• $$|A^T| = |A|$$ (transpose does not change the determinant)
• If any two rows (or columns) are interchanged, the determinant changes sign.
• If any two rows (or columns) are identical, $$|A| = 0$$.
• If all elements of a row are multiplied by $$k$$, the determinant is multiplied by $$k$$.
• $$|kA| = k^n |A|$$ for an $$n \times n$$ matrix.
• Adding a multiple of one row to another does not change the determinant.
• $$|AB| = |A| \cdot |B|$$

Cofactor Expansion

The determinant can be computed along any row or column:

$$$|A| = \sum_{j=1}^{n} a_{ij} C_{ij} \quad \text{(expansion along row } i\text{)}$$$

Properties of Adjoint

• $$A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| \cdot I$$
• $$|\text{adj}(A)| = |A|^{n-1}$$ for an $$n \times n$$ matrix
• $$\text{adj}(AB) = \text{adj}(B) \cdot \text{adj}(A)$$
• $$\text{adj}(A^T) = (\text{adj}(A))^T$$

Inverse of a Matrix

$$$A^{-1} = \frac{1}{|A|} \text{adj}(A) \qquad (|A| \neq 0)$$$

$$2 \times 2$$ Inverse

For $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ with $$ad - bc \neq 0$$:

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

Properties of Inverse

• $$(A^{-1})^{-1} = A$$
• $$(AB)^{-1} = B^{-1}A^{-1}$$ (reverse order!)
• $$(A^T)^{-1} = (A^{-1})^T$$
• $$|A^{-1}| = \dfrac{1}{|A|}$$
• $$(kA)^{-1} = \dfrac{1}{k}A^{-1}$$

For the system $$AX = B$$ where $$A$$ is $$n \times n$$ and $$|A| \neq 0$$:

$$$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}$$$

• $$D = |A|$$ = determinant of the coefficient matrix
• $$D_x$$ = determinant obtained by replacing the $$x$$-column with $$B$$
• $$D_y$$ = determinant obtained by replacing the $$y$$-column with $$B$$
• $$D_z$$ = determinant obtained by replacing the $$z$$-column with $$B$$

Solving $$AX = B$$ Using the Inverse

If $$|A| \neq 0$$:

$$$X = A^{-1}B$$$

Classification of Systems $$AX = B$$

Let $$D = |A|$$:

Case 1: $$D \neq 0$$ (Non-singular)

• The system has a unique solution (consistent).

Case 2: $$D = 0$$ (Singular)

• If $$(\text{adj}\,A) \cdot B = O$$: infinitely many solutions (consistent).
• If $$(\text{adj}\,A) \cdot B \neq O$$: no solution (inconsistent).

Homogeneous System $$AX = O$$

A homogeneous system always has at least the trivial solution $$X = O$$.

• If $$|A| \neq 0$$: only the trivial solution $$x = y = z = 0$$.
• If $$|A| = 0$$: infinitely many (non-trivial) solutions exist.

Key Formulas at a Glance