Unit Summary

This section pulls together the core definitions, properties and formulae of the Algebra unit of CBSE Class 12 Applied Mathematics (subject code 241) — matrices, their operations, determinants and their applications — as a single quick-revision recap for board-exam and previous-year-paper practice.

Matrices and their types

• Matrix: A rectangular arrangement of numbers, symbols, expressions or functions set out in rows and columns is a matrix (plural matrices); each entry in the arrangement is an element of the matrix.
• Notation and order: A matrix is named with a capital letter, $A = [a_{ij}]$, where $a_{ij}$ is the element in the $i$th row and $j$th column, with $1 \le i \le m$ and $1 \le j \le n$. If $A$ has $m$ rows and $n$ columns, the expression $m \times n$ is its order. The entries $a_{ij}$ with $i = j$ (that is $a_{11}, a_{22}, a_{33}, \dots$) form the diagonal; entries with $i \ne j$ are the non-diagonal elements.
• Rectangular matrix: the number of rows is not equal to the number of columns.
• Square matrix: the number of rows equals the number of columns.
• Row matrix: exactly one row. Column matrix: exactly one column.
• Diagonal matrix: a square matrix whose non-diagonal entries are all zero, i.e. $a_{ij} = 0$ for all $i \ne j$.
• Scalar matrix: a diagonal matrix whose diagonal entries are all equal, i.e. $a_{ij} = k$ for all $i = j$, with $k \ne 0$.
• Identity (unit) matrix: a scalar matrix whose diagonal entries are all $1$, i.e. $a_{ij} = 1$ for all $i = j$; denoted $I$.
• Zero matrix: every element is zero, i.e. $a_{ij} = 0$; denoted $O$.
• Equal matrices: two matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ of the same order $m \times n$ are equal when each corresponding element agrees, i.e. $a_{ij} = b_{ij}$ for every $1 \le i \le m,\ 1 \le j \le n$.

Algebra of matrices

• Scalar multiplication: for a scalar $k$, $kA = k[a_{ij}] = [k\,a_{ij}]$ — every element is multiplied by $k$. Scalar multiplication distributes both ways: $k(A + B) = kA + kB$ and $(k + p)A = kA + pA$.
• Negative of a matrix: for a non-zero matrix $A$, the matrix $-A$ of the same order satisfies $A + (-A) = O$, where $O$ is the zero matrix; it is the additive inverse of $A$.
• Addition: for $A = [a_{ij}]$ and $B = [b_{ij}]$ of the same order, the sum is $S = A + B = [s_{ij}]$ with $s_{ij} = a_{ij} + b_{ij}$.
• Properties of addition: matrices can be added only when they share the same order, and the result keeps that order. Addition is commutative ($A + B = B + A$) and associative ($A + (B + C) = (A + B) + C$); the zero matrix $O$ is the additive identity ($A + O = O + A = A$); and $-A$ is the additive inverse ($A + (-A) = O$).
• Subtraction: the difference is $D = A - B = [d_{ij}]$ with $d_{ij} = a_{ij} - b_{ij}$ (same order).
• Multiplication: for $A = [a_{ij}]$ of order $m \times n$ and $B = [b_{jk}]$ of order $n \times p$, the product $P = AB = [p_{ik}]$ has order $m \times p$; each entry pairs the $i$th row of $A$ with the $k$th column of $B$: $p_{ik} = a_{i1}b_{1k} + a_{i2}b_{2k} + a_{i3}b_{3k} + \dots + a_{in}b_{nk}$.
• Properties of multiplication: matrix multiplication is associative, $(AB)C = A(BC)$, and distributive over addition and subtraction, $A(B + C) = AB + AC$ and $A(B - C) = AB - AC$, whenever the products are defined.
• Multiplicative identity: for a square matrix $A$ of order $m \times m$ there is an identity matrix $I$ of the same order with $IA = AI = A$.

Transpose, symmetric and skew-symmetric matrices

• Transpose: interchanging the rows and columns of $A$ (order $m \times n$) gives its transpose $A'$ of order $n \times m$.
• Transpose properties: $(A')' = A$; $(kA)' = kA'$ for any constant $k$; $(A + B)' = A' + B'$; and $(AB)' = B'A'$.
• Symmetric matrix: a square matrix with $A' = A$. Skew-symmetric matrix: a square matrix with $A' = -A$.
• For any square matrix $A$ with real entries, $A + A'$ is a symmetric matrix and $A - A'$ is a skew-symmetric matrix.

Determinants

• Determinant: a rule that assigns to each square matrix a single number $|A|$ (real or complex) — formally a function $f$ with $f(A) = k$.
• Minor: the minor $M_{ij}$ of element $a_{ij}$ is the determinant left after deleting the $i$th row and $j$th column in which $a_{ij}$ lies.
• Cofactor: the cofactor of $a_{ij}$, written $A_{ij}$ or $C_{ij}$, is $A_{ij} = (-1)^{i+j} M_{ij}$, where $M_{ij}$ is the minor of $a_{ij}$.
• Adjoint: the adjoint $\operatorname{adj} A$ is the transpose of the matrix of cofactors of the square matrix $A$.
• Adjoint properties: for a square matrix $A$ of order $n$, $A(\operatorname{adj} A) = (\operatorname{adj} A)\,A = |A|\,I$ (with $I$ the identity of order $n$), and $|\operatorname{adj}(A)| = |A|^{\,n-1}$.
• Singular / non-singular: $A$ is singular when $|A| = 0$ and non-singular when $|A| \ne 0$. …