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# IIT JEE Maths Matrices and Determinants

## Matrices and Determinants PDF Notes, Important Questions and Synopsis

SYNOPSIS

1. Determinants
Every square matrix can be associated to a number called a Determinant.
For A  → Square matrix; |A| or det A or D → denotes the determinant of A

2. Sub-matrix
A matrix obtained after deleting some rows or columns is called a sub-matrix.

3. Minor & Cofactor
A minor is the determinant of the sub-matrix obained by deleting the ith row and jth column.
It is denoted by Mij.
A cofactor is denoted by Cij, and it is given by Cij = (-1)i+j Mij.

4. Finding Determinant
A matrix should be a square matrix of order greater than 1, A = [aij]nxn.
A determinant of a matrix A is defined as the sum of the products of elements of any one row (or column) with corresponding cofact

|A|= a11C11 + a12C12 + a13C13 (Using first row)ors.

5. Properties of a Determinant

1. The value of a determinant remains the same if the rows and columns are interchanged.
2. The value of a determinant changes in sign only if any two rows (or columns) of a determinant are interchanged.
3. If a determinant has any two rows (or columns) identical, then the value of the determinant is zero.

4. If all the elements of any row (or column) are multiplied by the same number, then the determinant is multiplied by that number.

5. If each element of any row (or column) can be expressed as a sum of two terms, then the determinant can be expressed as the sum of two determinants.

6. The value of a determinant does not change by adding to the elements of any row (or columns) the same multiples of the corresponding elements of any other row.

Note: While applying this property, at least one row or column must remain unchanged.

6. Cramer’s Rule (Determinant)
Consider three simultaneous linear equations:
a1x + b1y + c1z = d1 ,  a2x + b2y + c2z = d2  and  a3x + b3y + c3z = d3
The solution of the above system of linear equations is :

Matrices

A matrix is a rectangular array/arrangement of numbers along rows and columns.

Note: A matrix A = [aij], where aij is an element of the ith row and jth column.
Row number and column number are the same for diagonal elements.

1. Classification of Matrices

 Row Matrix                           A = [3   6   -8] Matrix having only                       OR one row.                                A = [1   2   6  0] Column MatrixMatrix having only one column. Square MatrixMatrix having same number of rows and columns. Zero (Null) MatrixMatrix having all elements equal to zero.Matrix having all elements equal to zero. Upper Triangular MatrixAll entries below the main diagonal are zero.All entries below the main diagonal are zero. Lower Triangular MatrixAll entries above the main diagonal are zero. Diagonal MatrixAll entries above and below the diagonal are zero.All entries above and below the diagonal are zero. Identity (Unit) MatrixAll diagonal entries are one and the rest are zero.All diagonal entries are one and the rest are zero.
2. Operations on Matrices

 Addition of MatricesOrder of the matrices must be same. Subtraction of MatricesOrder of the matrices must be same. Equality of matricesMatrices having same order with each of their elements equal. Transpose of a MatrixMatrix obtained after turning rows into columns and vice versa; denoted by AT.Matrix obtained after turning rows into columns and vice versa; denoted by AT.
3. Multiplication of Matrices

 Multiplication of a matrix with a scalar Each element of the matrix is multiplied by the scalar. Multiplication of two matricesLet A and B be two matrices, then A Let A and B be two matrices, then A ´ B is possible only if ‘No. of columns of 1st matrix = No. of rows of 2nd matrix’. Process of Multiplication of two Matrices
4. Inverse of a Matrix

Inverse of a matrix A is denoted by A-1 such that (matrix)  (inverse of matrix) = I
i.e. A × A-1 = I or A-1 × A = I. But A × A-1 ≠ A-1 ´ A.
Steps of finding inverse of a matrix:

 Step I Check whether matrix A is singular or non-singular, i.e. |A| = 0 Þ Singular |A|   ≠ 0 Þ Non-singular Step II If matrix A is non-singular, then find the value of determinant of A and also find the adjoint matrix A. Step III Use the formula

Note: if A is non-singular.
If A = diag (a11, a22, …., ann), then
5. Type of Square matrices

 Nilpotent Matrix If for some least +ve integer p, then Ap =0 is a nilpotent matrix. Idempotent Matrix If A2= A then A is an idempotent matrix. Symmetric Matrix   If AT = A, then A is a symmetric matrix. Skew-symmetric Matrix If AT = -A, then A is a skew-symmetric matrix. Also, all the diagonal elements are zero. Involutory Matrix  If A2 = I, then A is an involutory matrix. Unitary Matrix If A’(A’)T = I, where A’ is the complex conjugate of A,   then A is a unitary matrix. Orthogonal Matrix  If A is a square matrix such that ATA = I = ATA or AT = A-1, then A is an orthogonal matrix.

6. For any square matrix A, A + AT is symmetric and A - AT is skew-symmetric.
7. Every square matrix can be expressed as a sum of symmetric and skew-symmetric matrices.

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