Matrices and Determinants for IIT-JEE and AIEEE - Quick Revision
A matrix is defined as a rectangular array of elements. |
If the arrangement has m rows and n columns, then the matrix is of order mxn (read as m by n). |
A matrix is enclosed by a pair of parameters such as ( ) or [ ]. It is denoted by a capital letter. |
Two matrices are said to be comparable if they have the same order. |
Addition and subtraction of two matrices is possible only if they have the same order. |
If two matrices A and B are of same order, then A - B = A + (- B). |
Commutative law, associative law holds good for addition of matrices. |
The additive identity of a matrix A of order mxn is the zero matrix of order mxn. |
The additive inverse of a matrix A is -A. |
The multiplication of two matrices A and B is possible if the number of columns of A is equal to the number of rows B. |
Suppose A is a matrix of order mxn and B is a matrix of order nxp, the matrix AB is of order mxp. |
If A, B and C are the matrices which can be multiplied then |
(a) Matrix multiplication is not commutative, |
i.e., AB BA (always) |
(b) Associative law holds good for matrix multiplication, |
i.e., (AB)C = A(BC) |
(c) Matrix multiplication is distributive with respect to addition |
A(B + C) = AB + AC |
or (A + B)C = AC + BC |
If A is a matrix of order mxn and is a scalar (real or complex) then the matrix kA is obtained by multiplying each element of A by k. |
To every square matrix, a value can be associated which is known as the determinant of the matrix. |
Note that the determinant of kA where k is a scalar and A is a square matrix, is given by k^{n} times determinant of A. |
i.e., is |kA| = k^{n} |A| |
The value of the determinant remain unchanged if its rows and columns are interchanged |
If two rows or columns of a determinant are interchanged, then the sign of the determinant is changed. |
If any two rows or columns of a determinant are equal, then its value is zero. |
If each element of a row or column of a determinant multiplied by k, then its value is multiplied by k. |
If two rows or columns of determinant are proportional, the value of the determinant is zero. |
A square A = [a_{i}_{j}] is said to be symmetric if A^{T} = A, i.e., if |
a_{i}_{j} = a_{j}_{i} |
A square matrix A is said skew symmetric if A^{T} = - A, |
i.e., a_{i}_{j} = - a_{j}_{i} |
Any square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix as follows |
For a 2 x 2 matrix, the adjoint is got by interchanging elements in the leading diagonal and changing signs in the other diagonal. |
If A =[a_{i}_{j}]_{m}_{x}_{n} is a matrix of order mxn. The minor of a_{i}_{j} of |A|, denoted by M_{i}_{j}, is given by the determinant which is obtained by deleting i^{t}^{h} row j^{t}^{h} column of |A|. |
The co-factor of the determinant of the A = [a^{i}^{j}]_{m}_{x}_{n}, denoted by A_{i}_{j} is given by |
A_{i}_{j} = (-1)^{i}^{+}^{j} M_{i}_{j} |
The transpose of a matrix A, denoted by A^{T}, is obtained by interchanging the rows and columns of A. |
The adjoint of a square matrix A = [a_{i}_{j}] is defined as the transpose of the matrix [A_{i}_{j}] where A_{i}_{j} is the co-factor of the element a_{i}_{j}. |
Adjoint of A is denoted by Adj A. |
Note that the concept of adj is only for square matrix. |
A square matrix A is said to be non-singular if |A| 0. |
Let A be a square matrix of order n. If there exists a square matrix B of order n, such that AB = BA = I_{n}, where I_{n} is the identify matrix of order n, then B is called the inverse of A. |
The inverse of a matrix A exists if and only if |A| 0. |
In other words, every non-singular matrix is invertible. |
The area of a triangle whose vertices are (x_{1}, y_{1}), (x_{2}, y_{2}) and |
The following are the steps to solve a system of linear equations |
using Cramer's rule. |
Step 1: Find the value of the determinant |
Step 2: If D 0, then the system has unique solution, given by |
Where D_{1}, D_{2} and D_{3} are the determinants obtained from D by replacing respectively the first column, 2^{n}^{d} column and third column containing the constant terms d_{1}, d_{2}, d_{3}. |
Step 3: If D = 0, the system may have infinite number of solutions or no solution. |
A system of linear equations is said to be consistent if it has at least are solution, otherwise it is inconsistent. |
Let A be asquare matrix of order n. Following are the steps to find the inverse of a matrix. |
Step 1: Find the value of the determinants A. That is, find |A| |
Step 2: If |A| = 0, inverse of the matrix A does not exists. |
Step 3: If |A| 0, find the co-factors A_{i}_{j} of all the elements of A. |
Step 4: Find adj A, the transpose of the matrix of co-factors A_{i}_{j}. |
Step 5: |
Following are the steps to solve a system of linear equations with three unknown, using inverse of a matrix (Matrix method) |
Let the given system of equations be |
Step 1: |
The system of linear equations may be expressed as AX = B. |
Step 2: Find |A|. If |A| 0, the system has unique solution which is given by X = A^{-}^{1}B. |
Step 3: |
If |A| = 0, put x = k (y = k or z = k) in any two of the given equations and find y and z in terms of k. |
Substitute these values of x, y and z in terms of k in the third equation. If the third equation is satisfied by these values of x, y and z, then the system has infinitely many solutions. |
If the third equation is not satisfied, the system has no solution. |