Matrices QRT – Class 12 Maths (Free Download)

Matrices QRT – Quick Revision Tables bring the important concepts, formulas and results of the chapter together in one place. Use these tables for a quick recap while revising, practising questions or preparing for your exams.

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Table 1: Matrices QRT – Basics

Concept	Formula / Result
Matrix	An ordered rectangular array of numbers or functions. The numbers or functions are called the elements or entries of the matrix.
Matrix Notation	Matrices are denoted by capital letters, e.g. \displaystyle A=[a_{ij}]_{m\times n}.
Elements Considered	In this chapter, matrix elements are real numbers or functions taking real values.
Order of a Matrix	A matrix with \displaystyle m rows and \displaystyle n columns has order \displaystyle m\times n.
Number of Elements	An \displaystyle m\times n matrix has \displaystyle mn elements.
Row	Horizontal arrangement of elements.
Column	Vertical arrangement of elements.
Element \displaystyle a_{ij}	The \displaystyle (i,j)th element lies in the \displaystyle ith row and \displaystyle jth column.
Values of \displaystyle i,j	\displaystyle 1\leq i\leq m,\; 1\leq j\leq n; where i,j\in\mathbb N

Maths Better Tip… If a matrix has a fixed number of elements, its possible orders are obtained from the factor pairs of that number. For example, a matrix with 8 elements can have orders \displaystyle 1\times8,\;8\times1,\;2\times4,\;4\times2.

QRT 2: Types of Matrices

Type of Matrix	Quick Condition / Form
Row Matrix	Has only one row; order \displaystyle 1\times n.
Column Matrix	Has only one column; order \displaystyle m\times1.
Rectangular Matrix	Number of rows is not equal to the number of columns; \displaystyle m\neq n.
Square Matrix	Number of rows is equal to the number of columns; \displaystyle m=n. A square matrix with \displaystyle n rows and \displaystyle n columns is of order \displaystyle n.
Main Diagonal	In a square matrix \displaystyle A=[a_{ij}]_{n\times n}, the elements \displaystyle a_{11},a_{22},\ldots,a_{nn} constitute the main diagonal.
Diagonal Matrix	A square matrix in which all non-diagonal elements are zero; \displaystyle a_{ij}=0,\quad i\neq j.
Scalar Matrix	A diagonal matrix whose diagonal elements are all equal to the same constant \displaystyle k: \displaystyle a_{ij}=\begin{cases}k,&i=j\\0,&i\neq j\end{cases}
Identity Matrix	A square matrix with 1 on the main diagonal and 0 elsewhere; \displaystyle I_n=[a_{ij}],\quad a_{ij}=\begin{cases}1,&i=j\\0,&i\neq j\end{cases}
Zero Matrix	A matrix in which every element is zero; denoted by \displaystyle O. Its order is understood from the context.

Maths Better Tip… Remember the hierarchy: every identity matrix is a scalar matrix, every scalar matrix is a diagonal matrix, and every diagonal matrix is a square matrix. A scalar matrix becomes an identity matrix when \displaystyle k=1

\displaystyle I_n\;\Rightarrow\;\text{Scalar}\;\Rightarrow\;\text{Diagonal}\;\Rightarrow\;\text{Square}

The reverse is not generally true.

Table 3: Equality of Matrices QRT

Concept	Formula / Result
Equality of Matrices	Two matrices \displaystyle A and \displaystyle B are equal if they have the same order and all corresponding elements are equal.
Same Order	\displaystyle A and \displaystyle B must have the same number of rows and columns.
Corresponding Elements	\displaystyle a_{ij}=b_{ij} for all possible values of \displaystyle i and \displaystyle j.
Symbolic Form	\displaystyle A=B if and only if A and B have the same order and \displaystyle a_{ij}=b_{ij} for all \displaystyle i,j.
Finding Unknowns	When two matrices are given as equal, compare their corresponding elements and solve the resulting equations to find the unknowns.
Different Orders	Matrices of different orders can never be equal.

Maths Better Tip… When two matrices are given as equal, don’t compare the matrices as a whole. Compare their corresponding elements one by one and solve the resulting equations.

Matrices QRT 4: Addition, Scalar Multiplication & Difference

Operation / Property	Formula / Result
Addition	Defined only when A and B have the same order. \displaystyle A+B=[a_{ij}+b_{ij}]
Scalar Multiplication	If k is a scalar, multiply every element of A by k: \displaystyle kA=[ka_{ij}]
Negative of a Matrix	\displaystyle -A=(-1)A=[-a_{ij}]
Difference	Defined only when A and B have the same order. \displaystyle A-B=A+(-B)=[a_{ij}-b_{ij}]
Commutative Law	\displaystyle A+B=B+A
Associative Law	\displaystyle (A+B)+C=A+(B+C)
Additive Identity	\displaystyle A+O=O+A=A
Additive Inverse	\displaystyle A+(-A)=(-A)+A=O
Scalar Distributive Law	\displaystyle k(A+B)=kA+kB
Scalar Addition Law	\displaystyle (k+l)A=kA+lA
Scalar 1	\displaystyle 1A=A
Scalar 0	\displaystyle 0A=O

Maths Better Tip… For addition and subtraction of matrices, always check the order first. If the orders are different, the operation is not defined.

\displaystyle A_{m\times n}+B_{p\times q}\text{ is defined only when }m=p\text{ and }n=q.

Maths Better Tip… For addition, add corresponding elements. For subtraction, subtract corresponding elements, and for scalar multiplication, multiply every element by the scalar.

Table 5: Multiplication of Matrices QRT

Concept	Formula / Result
Condition for Multiplication	\displaystyle A_{m\times n}B_{n\times p} is defined when the number of columns of A = number of rows of B.
Order of Product	\displaystyle AB is of order \displaystyle m\times p. Inner dimensions match; outer dimensions give the order.
Element of Product	If \displaystyle AB=C, then \displaystyle c_{ik}=\sum_{j=1}^{n}a_{ij}b_{jk}.
Row × Column Rule	Each element of \displaystyle AB is obtained by multiplying the corresponding elements of a row of A and a column of B, then adding the products.
Both AB and BA Defined	If \displaystyle A_{m\times n} and \displaystyle B_{k\times l}, both \displaystyle AB and \displaystyle BA are defined iff \displaystyle n=k and \displaystyle l=m.
AB may exist, BA may not	Even if \displaystyle AB is defined, \displaystyle BA need not be defined.
Commutativity	Matrix multiplication is not commutative in general: \displaystyle AB\neq BA.
Same Order ≠ Same Product	Even when \displaystyle AB and \displaystyle BA are defined and have the same order, they need not be equal.
Zero Product	\displaystyle AB=O does not necessarily mean \displaystyle A=O or \displaystyle B=O.

Maths Better Tip… For matrix multiplication, remember: the inner dimensions must match, and the outer dimensions give the order of the product.

\displaystyle (m\times n)(n\times p)=m\times p

Caution… Even when both products are defined and have the same order, you cannot conclude that they are equal. Matrix multiplication is not commutative in general.

\displaystyle AB\neq BA\quad\text{in general}

Special Case: Diagonal matrices of the same order commute.

\displaystyle AB=BA

View or Download the free Matrices QRT – Class 12 Maths PDF for quick revision anytime.

Table 6: Properties of Matrix Multiplication

Property	Formula / Result
Associative Law	\displaystyle (AB)C=A(BC) whenever both sides are defined.
Left Distributive Law	\displaystyle A(B+C)=AB+AC whenever both sides are defined.
Right Distributive Law	\displaystyle (A+B)C=AC+BC whenever both sides are defined.
Multiplicative Identity	For a square matrix A, the identity matrix I of the same order satisfies \displaystyle AI=IA=A.

Maths Better Tip… Do not confuse associative and commutative properties. Associative changes the grouping, while commutative changes the order. Matrix multiplication is associative but not commutative in general.

\displaystyle (AB)C=A(BC)\quad\text{but generally}\quad AB\neq BA

Matrices QRT 7: Transpose of a Matrix

Concept	Formula / Result
Notation	The transpose of \displaystyle A is denoted by \displaystyle A' or \displaystyle A^T.
Transpose	If \displaystyle A=[a_{ij}]_{m\times n}, its transpose is obtained by interchanging rows and columns: \displaystyle A'=A^T=[a_{ji}]_{n\times m}
Order of Transpose	An \displaystyle m\times n matrix becomes an \displaystyle n\times m matrix after taking transpose.
Square Matrix	A square matrix remains square and of the same order after taking transpose.
Transpose of Transpose	\displaystyle (A')'=A
Transpose of Scalar Multiple	\displaystyle (kA)'=kA'
Transpose of Sum or Difference	\displaystyle (A±B)'=A'±B'
Transpose of Product (Reversal Law)	\displaystyle (AB)'=B'A'

Maths Better Tip… When taking the transpose of a product, the order reverses. Do not forget to reverse the matrices.

\displaystyle (AB)'=B'A'

Table 8: Symmetric & Skew-Symmetric Matrices QRT

Feature	Symmetric Matrix	Skew-Symmetric Matrix
Condition	\displaystyle A'=A	\displaystyle A'=-A
Element-wise Condition	\displaystyle a_{ij}=a_{ji}	\displaystyle a_{ij}=-a_{ji} and \displaystyle a_{ii}=0
Order	Must be a square matrix.	Must be a square matrix.
Diagonal Elements	Can be any real numbers.	All diagonal elements are zero.
Both Symmetric & Skew-Symmetric	A matrix that is both symmetric and skew-symmetric is the zero matrix.

Important Result	Result
Sum with Transpose	\displaystyle A+A' is always symmetric.
Difference with Transpose	\displaystyle A-A' is always skew-symmetric.
Symmetric Part	\displaystyle \frac{1}{2}(A+A') is symmetric.
Skew-Symmetric Part	\displaystyle \frac{1}{2}(A-A') is skew-symmetric.
Decomposition	Every square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix: \displaystyle A=\frac{1}{2}(A+A')+\frac{1}{2}(A-A')

Important Result… If A and B are symmetric matrices of the same order, then AB is symmetric if and only if A and B commute.

\displaystyle AB\text{ is symmetric}\;\Longleftrightarrow\;AB=BA

Important Result… If A and B are symmetric matrices, then AB − BA is skew-symmetric.

\displaystyle (AB-BA)'=-(AB-BA)

Maths Better Tip… For a skew-symmetric matrix, first check the main diagonal. Every diagonal element must be zero. Also remember that a matrix which is both symmetric and skew-symmetric can only be the zero matrix.

QRT 9: Invertible Matrices & Inverse

Concept	Formula / Result
Invertible Matrix	A square matrix A is invertible if there exists a matrix B of the same order such that \displaystyle AB=BA=I.
Inverse	If \displaystyle AB=BA=I, then B is the inverse of A and is denoted by \displaystyle A^{-1}.
Square Matrix	Only a square matrix can have an inverse. A rectangular matrix does not possess an inverse.
Inverse is Unique	If the inverse of a square matrix exists, it is unique.
Inverse of an Inverse	\displaystyle (A^{-1})^{-1}=A
Inverse of a Product	If A and B are invertible matrices of the same order, then \displaystyle (AB)^{-1}=B^{-1}A^{-1}.

Maths Better Tip… Remember the order reversal for both transpose and inverse of a product.

\displaystyle (AB)'=B'A'\quad\text{and}\quad (AB)^{-1}=B^{-1}A^{-1}

Matrices QRT 10: Important Results & Quick Recall

Important Result	Formula / Result
Both Symmetric & Skew-Symmetric	A matrix which is both symmetric and skew-symmetric is the zero matrix.
Symmetric Matrices	If A and B are symmetric matrices of the same order, then \displaystyle AB\text{ is symmetric}\iff AB=BA.
Difference of Products	If A and B are symmetric, then \displaystyle AB-BA is skew-symmetric.
Congruence-Type Result	If A is symmetric, then \displaystyle B'AB is symmetric. If A is skew-symmetric, then \displaystyle B'AB is skew-symmetric.
Inverse Condition	A and B are inverses of each other only when \displaystyle AB=BA=I.

Quick Recall	Result / Meaning
Matrix Square	\displaystyle A^2=AA, defined when A is square.
Matrix Cube	\displaystyle A^3=A^2A=AAA, defined when A is square.
Polynomial in A	Expressions such as \displaystyle A^2-5A+7I are formed by matrix addition and multiplication.
Matrix Equation	For an equation involving matrices, compare corresponding elements after carrying out the required matrix operations.

Maths Better Tip… For quick revision, don’t just memorise formulas. Remember the conditions under which they are valid, the important exceptions and the common traps.

Quick Questions — Test Yourself

Maths Better Tip… Try to answer each question before opening it. Then click the arrow to reveal the answer and check yourself.

Can two non-zero matrices have a zero product?

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Yes. \displaystyle AB=O does not necessarily mean \displaystyle A=O or \displaystyle B=O.

Can \displaystyle AB be defined while \displaystyle BA is not?

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Yes. Matrix multiplication depends on the dimensions of the matrices, so \displaystyle AB may be defined even when \displaystyle BA is not.

If both \displaystyle AB and \displaystyle BA are defined, must they be equal?

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No. Even when both products are defined, matrix multiplication is not commutative in general: \displaystyle AB\neq BA.

Can a matrix be both symmetric and skew-symmetric?

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Yes, but only if it is the zero matrix. If \displaystyle A is both symmetric and skew-symmetric, then \displaystyle A=A'=-A, so \displaystyle A=O.

If \displaystyle A and \displaystyle B are symmetric matrices, must \displaystyle AB be symmetric or skew-symmetric?

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No. In general, \displaystyle AB can be neither symmetric nor skew-symmetric. Since \displaystyle (AB)'=BA, the product is symmetric if \displaystyle AB=BA, and skew-symmetric if \displaystyle AB=-BA.

Can a rectangular matrix have an inverse?

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No.

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