Matrices (Class 12 NCERT)

Cheatsheet Content

### Introduction to Matrices - A matrix is an ordered rectangular array of numbers or functions. - The numbers or functions are called the elements or entries of the matrix. - **Order of a matrix:** A matrix having $m$ rows and $n$ columns is called a matrix of order $m \times n$. - Example: $A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix}$ is a $2 \times 3$ matrix. - The element $a_{ij}$ refers to the element in the $i$-th row and $j$-th column. ### Types of Matrices - **Column Matrix:** A matrix having only one column. Order $m \times 1$. - **Row Matrix:** A matrix having only one row. Order $1 \times n$. - **Square Matrix:** A matrix in which the number of rows is equal to the number of columns ($m=n$). - **Diagonal Matrix:** A square matrix where all non-diagonal elements are zero. $a_{ij}=0$ for $i \neq j$. - **Scalar Matrix:** A diagonal matrix where all diagonal elements are equal. $a_{ij}=0$ for $i \neq j$ and $a_{ii}=k$ for some scalar $k$. - **Identity Matrix ($I$):** A scalar matrix where diagonal elements are 1. $a_{ii}=1$ and $a_{ij}=0$ for $i \neq j$. - **Zero Matrix (Null Matrix):** A matrix in which all elements are zero. Denoted by $O$. ### Equality of Matrices - Two matrices $A$ and $B$ are said to be equal if: 1. They are of the same order. 2. Each element of $A$ is equal to the corresponding element of $B$. - $a_{ij} = b_{ij}$ for all $i, j$. ### Operations on Matrices #### Addition of Matrices - Two matrices can be added if they are of the same order. - If $A = [a_{ij}]$ and $B = [b_{ij}]$ are two $m \times n$ matrices, then $A+B = [a_{ij} + b_{ij}]$. - **Properties of Matrix Addition:** - Commutative: $A+B = B+A$ - Associative: $(A+B)+C = A+(B+C)$ - Existence of Additive Identity: $A+O = O+A = A$ (where $O$ is zero matrix of same order as $A$) - Existence of Additive Inverse: $A+(-A) = O$ #### Scalar Multiplication - If $A = [a_{ij}]$ is a matrix and $k$ is a scalar, then $kA = [k a_{ij}]$. - **Properties of Scalar Multiplication:** - $k(A+B) = kA + kB$ - $(k+l)A = kA + lA$ #### Matrix Multiplication - Two matrices $A$ and $B$ can be multiplied if the number of columns in $A$ is equal to the number of rows in $B$. - If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then $AB$ is an $m \times p$ matrix. - $(AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}$ - **Properties of Matrix Multiplication:** - Not Commutative in general: $AB \neq BA$. $AB$ and $BA$ may not even be defined or of the same order. - Associative: $(AB)C = A(BC)$ - Distributive: $A(B+C) = AB + AC$ and $(A+B)C = AC + BC$ - Existence of Multiplicative Identity: $AI = IA = A$ (where $I$ is identity matrix of appropriate order) - If $AB=O$, it does not necessarily imply $A=O$ or $B=O$. ### Transpose of a Matrix - The transpose of a matrix $A$ (denoted by $A'$ or $A^T$) is obtained by interchanging its rows and columns. - If $A = [a_{ij}]$ is an $m \times n$ matrix, then $A' = [a_{ji}]$ is an $n \times m$ matrix. - **Properties of Transpose:** - $(A')' = A$ - $(kA)' = kA'$ (where $k$ is any scalar) - $(A+B)' = A' + B'$ - $(AB)' = B'A'$ (Reversal Law) ### Symmetric and Skew-Symmetric Matrices - **Symmetric Matrix:** A square matrix $A$ is symmetric if $A' = A$. - This means $a_{ij} = a_{ji}$ for all $i,j$. - **Skew-Symmetric Matrix:** A square matrix $A$ is skew-symmetric if $A' = -A$. - This means $a_{ij} = -a_{ji}$ for all $i,j$. - For a skew-symmetric matrix, all diagonal elements are zero ($a_{ii}=0$). - **Theorem:** Any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix. - $A = \frac{1}{2}(A+A') + \frac{1}{2}(A-A')$ - Here, $\frac{1}{2}(A+A')$ is symmetric and $\frac{1}{2}(A-A')$ is skew-symmetric. ### Elementary Operations (Transformations) of a Matrix - There are six elementary operations (or transformations) on a matrix, three for rows and three for columns. 1. **Interchange of any two rows (or columns):** $R_i \leftrightarrow R_j$ (or $C_i \leftrightarrow C_j$) - Example: $R_1 \leftrightarrow R_2$ means swapping row 1 and row 2. 2. **Multiplication of the elements of any row (or column) by a non-zero number:** $R_i \rightarrow kR_i$ (or $C_i \rightarrow kC_i$) - Example: $R_1 \rightarrow 2R_1$ means multiplying all elements of row 1 by 2. 3. **Addition to the elements of any row (or column) the corresponding elements of any other row (or column) multiplied by a non-zero number:** $R_i \rightarrow R_i + kR_j$ (or $C_i \rightarrow C_i + kC_j$) - Example: $R_1 \rightarrow R_1 + 3R_2$ means adding 3 times elements of row 2 to corresponding elements of row 1. ### Invertible Matrices - If $A$ is a square matrix of order $m$, and if there exists another square matrix $B$ of the same order $m$, such that $AB = BA = I$ (where $I$ is the identity matrix of order $m$), then $B$ is called the inverse matrix of $A$. - The inverse of $A$ is denoted by $A^{-1}$. - An invertible matrix is also called a non-singular matrix. - **Uniqueness of Inverse:** The inverse of a square matrix, if it exists, is unique. - **Theorem:** If $A$ and $B$ are invertible matrices of the same order, then $(AB)^{-1} = B^{-1}A^{-1}$. - **Finding Inverse using Elementary Operations:** - To find $A^{-1}$ using row operations, write $A = IA$. Apply a sequence of row operations to both sides until $I = BA$ is obtained. Then $B=A^{-1}$. - To find $A^{-1}$ using column operations, write $A = AI$. Apply a sequence of column operations to both sides until $I = AB$ is obtained. Then $B=A^{-1}$. - If, during the process of reducing $A$ to $I$, we obtain all zeros in one or more rows (or columns) of the matrix on the LHS, then $A^{-1}$ does not exist. - **Note:** Never mix row and column operations when finding inverse. Stick to one type.