Unit 2: Algebra
Matrices, Determinants & System of Linear Equations
1. Introduction to Matrices
| Definition | A rectangular arrangement of numbers into rows and columns. Order: $$m \times n$$ (where $$m$$ is rows, $$n$$ is columns). |
| Types of Matrices |
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| Equality | Two matrices are equal if they have the same order and corresponding elements are identical ($$a_{ij} = b_{ij}$$). |
2. Transpose and Symmetric Matrices
| Transpose ($$A’$$ or $$A^T$$) | Obtained by interchanging rows and columns. If order of $$A$$ is $$m \times n$$, order of $$A’$$ is $$n \times m$$. |
| Symmetric Matrix | $$A’ = A$$ (i.e., $$a_{ij} = a_{ji}$$). |
| Skew-Symmetric Matrix | $$A’ = -A$$ (i.e., $$a_{ij} = -a_{ji}$$). Note: Diagonal elements are always zero. |
| Decomposition | $$A = \frac{1}{2}(A + A’) + \frac{1}{2}(A – A’)$$ (Sum of Symmetric and Skew-symmetric parts) |
3. Algebra of Matrices
| Addition/Subtraction | Possible only if matrices have the same order. Add/Subtract corresponding elements. |
| Matrix Multiplication ($$AB$$) | Possible if Columns of A = Rows of B. If $$A$$ is $$m \times n$$ and $$B$$ is $$n \times p$$, result $$AB$$ is $$m \times p$$. Note: $$AB \neq BA$$ usually (Not commutative). |
4. Determinants and Inverse
| Determinant ($$|A|$$) | Unique value associated with a square matrix. Singular: $$|A| = 0$$ (No Inverse). Non-Singular: $$|A| \neq 0$$ (Inverse exists). |
| Inverse ($$A^{-1}$$) | $$A^{-1} = \frac{1}{|A|} (\text{adj } A)$$ Where $$\text{adj } A$$ is the Adjoint Matrix. |
| Properties | 1. $$(AB)^{-1} = B^{-1}A^{-1}$$ 2. $$(A’)^{-1} = (A^{-1})’$$ 3. $$|AB| = |A||B|$$ |
5. Solving Simultaneous Equations
| Cramer’s Rule | $$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}$$ (Where $$D$$ is determinant of coefficients, and $$D_x, D_y, D_z$$ replace columns with constants). |
| Matrix Method | Write system as $$AX = B$$. Solution: $$X = A^{-1}B$$. |