Learning these TS Inter 1st Year Maths 1A Formulas Chapter 3 Matrices will help students to solve mathematical problems quickly.

## TS Inter 1st Year Maths 1A Matrices Formulas

→ Matrix: If the real or complex numbers are arranged in the form of a rectangular or square array consisting the complex numbers in horizontal and vertical lines, then that arrangement is called a matrix.

Ex: A = \(\left[\begin{array}{ccc}

1 & 2 & 4 \\

3 & 0 & -6

\end{array}\right]\), B = \(\left[\begin{array}{cc}

1 & 2 \\

4 & -3

\end{array}\right]\)

→ Order of a matrix: A matrix ‘A1 is said to be of type (or) order (or) size m × n (read as m by n), if the matrix A’ has m rows and n’ columns.

→ Square matrix: A matrix ‘A’ is said to be a square matrix if the number of rows in A is equal to the number of columns in A.

Ex: \(\left[\begin{array}{cc}

1 & -1 \\

0 & 4

\end{array}\right]\)_{2×2}

\(\left[\begin{array}{ccc}

2 & 0 & 1 \\

4 & -1 & 2 \\

7 & 6 & 9

\end{array}\right]\)_{2×2}

→ Trace of a matrix : If ‘A’ is a square matrix then the sum of elements in the principle diagonal of A’ is called trace of A’. It is denoted by tra A .

Ex: A = \(\left[\begin{array}{ccc}

2 & 0 & 1 \\

4 & -1 & 2 \\

7 & 6 & 9

\end{array}\right]\)

The elements of the principle diagonal = 2, – 1, 9

Tra (A) = 2 + (- 1) + 9 = 10.

→ Null matrix : A matrix ‘A’ is said to be a zero matrix or null matrix of every element of A is equal to zero.

Ex: O_{2} = \(\left[\begin{array}{ll}

0 & 0 \\

0 & 0

\end{array}\right]\)

O_{3×2} = \(\left[\begin{array}{ll}

0 & 0 \\

0 & 0 \\

0 & 0

\end{array}\right]\)

→ Upper triangular matrix: A square matrix A = [a_{ij}]_{n×n} is said to be an upper triangular matrix if a_{ij}, = 0, whenever i > i.

Ex: \(\left[\begin{array}{ccc}

2 & -1 & 5 \\

0 & 3 & 6 \\

0 & 0 & 1

\end{array}\right]_{3 \times 3}\)

→ Lower triangular matrix :

A square matrix A = [a_{ij}]_{n×n} is said to he a lower triangular matrix if a_{ij}, = 0 whenever i < j .

Ex: \(\left[\begin{array}{ccc}

2 & 0 & 0 \\

1 & 3 & 0 \\

5 & 4 & 6

\end{array}\right]_{3 \times 3}\)

→ Triangular matrix : A square matrix. A is said to be a triangular matrix ifA is either an upper triangular matrix or a lower triangular matrix.

Ex: \(\left[\begin{array}{ccc}

-1 & 0 & 0 \\

0 & 3 & 0 \\

7 & 5 & 2

\end{array}\right]_{3 \times 3}\)

→ Diagonal matrIx : A square matrix, A is said to be a diagonal matrix if A is both upper triangnlar and lower triangular matrix. (or) A square matrix in which every element is equal to zero except those of principle diagonal of the matrix Is a diagonal matrix.

Ex: \(\left[\begin{array}{ccc}

2 & 0 & 0 \\

0 & 3 & 0 \\

0 & 0 & -1

\end{array}\right]_{3 \times 3}\)

→ Scalar matrix : A diagonal matrix, A is said to be a scalar matrix if all elements in the principle diagonal are equal.

Ex: \(\left[\begin{array}{ll}

2 & 0 \\

0 & 2

\end{array}\right]_{2 \times 2}\)

\(\left[\begin{array}{lll}

3 & 0 & 0 \\

0 & 3 & 0 \\

0 & 0 & 3

\end{array}\right]_{3 \times 3}\)

→ Unit matrix : A diagonal matrix is said to be a unit matrix if every element in the principle diagonal is equal to unity. It is denoted by 1.

Ex: I_{2} = \(\left[\begin{array}{ll}

1 & 0 \\

0 & 1

\end{array}\right]\)

I_{3} = \(\left[\begin{array}{lll}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1

\end{array}\right]\)

→ Transpose of a matrix: The matrix obtained by changing the rows of a given matrix, A into columns is called transpose of ‘A’. It is denoted by A^{T} or A’.

Ex: A = \(\left[\begin{array}{ccc}

2 & 3 & -1 \\

1 & 2 & 3

\end{array}\right]\) then A^{T} = \(\left[\begin{array}{cc}

2 & 1 \\

3 & 2 \\

-1 & 3

\end{array}\right]\)

→ Symmetric matrix : A square matrix, A is said to be a symmetric matrix, if A^{T} = A.

Ex: If A = \(\left[\begin{array}{lll}

2 & 3 & 1 \\

3 & 4 & 5 \\

1 & 5 & 7

\end{array}\right]\), then A^{T} = \(\left[\begin{array}{lll}

2 & 3 & 1 \\

3 & 4 & 5 \\

1 & 5 & 7

\end{array}\right]^{\mathrm{T}}=\left[\begin{array}{lll}

2 & 3 & 1 \\

3 & 4 & 5 \\

1 & 5 & 7

\end{array}\right]\) = A

A is a symmetric matrix.

→ Skew symmetric matrix : A square matrix A’ is said to be a skew symmetric matrix, if A^{T} = – A.

Ex: A = \(\left[\begin{array}{ccc}

0 & 1 & -2 \\

-1 & 0 & 3 \\

2 & -3 & 0

\end{array}\right]\), then AT = \(\left[\begin{array}{ccc}

0 & 1 & -2 \\

-1 & 0 & 3 \\

2 & -3 & 0

\end{array}\right]^{\mathrm{T}}=\left[\begin{array}{ccc}

0 & -1 & 2 \\

1 & 0 & -3 \\

-2 & 3 & 0

\end{array}\right]=\left[\begin{array}{ccc}

0 & 1 & -2 \\

-1 & 0 & 3 \\

2 & -3 & 0

\end{array}\right]\) = -A

A is a skew symmetric matrix.

→ Adjoint of a matrix : The transpose of the matrix obtained by replacing the elements of a square matrix. A by the corresponding cofactors is called the adjoint matrix of A. It is denoted by Adi A or adj A.

→ Inverse of a square matrix : A sqiictre matrix A is said to be an invertible matrix, if there exists a square matrix, B such that AB = BA = I. The matrix B is called inverse of A’.

If A is a non-singular matrix, then A is invertible and A^{-1} = \(\frac{{adj} A}{{det} A}\)

→ Sub matrix : A matrix obtained by deleting some rows or columns or both of a matrix is called a sub matrix of the given matrix.

Ex: If A = \(\left[\begin{array}{cc}

1 & 2 \\

-1 & 2

\end{array}\right] \cdot\left[\begin{array}{lll}

1 & 2 & 3 \\

2 & 3 & 1

\end{array}\right] \cdot\left[\begin{array}{ll}

2 & 3 \\

3 & 1 \\

2 & 0

\end{array}\right]\) then some matrices of A are \(\left[\begin{array}{cc}

1 & 2 \\

-1 & 2

\end{array}\right] \cdot\left[\begin{array}{lll}

1 & 2 & 3 \\

2 & 3 & 1

\end{array}\right] \cdot\left[\begin{array}{ll}

2 & 3 \\

3 & 1 \\

2 & 0

\end{array}\right]\), [0]

→ Rank of a matrix : Let ‘A’ be a non zero matrix. The rank of A is defined as the maximum of the orders of the non singular square sub-matrices of A. The rank of a null matrix is defined ^ as zero. The rank of a matrix A is denoted by rank [A].

→ Rank of 3 × 3 matrix : Suppose A is a non-zero 3 × 3 matrix, then

- If A is a non – singular then its rank is 3.
- If A is a singular matrix and if at least one of its 2 × 2 sub matrix is non-singular, then the rank of A is 2.
- If A is a singular matrix and every 2×2, sub matrix is also singular, then the rank of ’A’ is 1.

→ Properties of matrices :

- If A and B are two matrices of same type, then A + B = B + A.
- If A, B and C are three matrices of same type then (A + B) + C = A + (B + C).
- If confirmability is assured for the matrices A, B and C then A(BC) = (AB)C.
- If confirmability is assured for the matrices A, B and C then

(a) A(B + C) = AB + AC

(b) (B + C) A = BA + CA. - If A is any matrix then (A
^{T})^{T}= A. - If A and B are two matrices of same type then (A + B)
^{T}= A^{T}+ B^{T}. - If A and B are two matrices for which confirmability for multiplication is assured then (AB)
^{T}= B^{T}. A^{T}. - If I is the identity matrix of order n then for every square matrix A of order n. AI = IA = A.
- If A is an invertible matrix then A
^{-1}is also invertible and (A^{-1})^{-1}= A. - If A and B are two invertible matrices of same type then AB is also invertible and (AB)
^{-1}= B^{-1}. A^{-1}. - If A is an invertible matrix then A is also invertible and (A
^{T})^{-1}= (A^{-1})^{T}

→ Methods of solving linear equations :

1. Cramer’s rule : Let a_{1}x + b_{1}y + c_{1}z = d_{1}

a_{2}x + b_{2}y + c_{2}z = d_{2}

a_{3}x + b_{3}y + c_{3}z = d_{3}

be a system of linear equations

2. Matrix inversion method : If ‘A’ is a non-singular matrix then the solution of AX’ = B is X = A^{-1} B.

3. Gauss Jordan method : Let a_{1}x + b_{1}y + c_{1}z = d_{1}

a_{2}x + b_{2}y + c_{2}z = d_{2}

a_{3}x + b_{3}y + c_{3}z = d_{3}

be a system of linear equations. If the augmented matrix \(\left[\begin{array}{llll}

\mathrm{a}_1 & \mathrm{~b}_1 & \mathrm{c}_1 & \mathrm{~d}_1 \\

\mathrm{a}_2 & \mathrm{~b}_2 & \mathrm{c}_2 & \mathrm{~d}_2 \\

\mathrm{a}_3 & \mathrm{~b}_3 & \mathrm{c}_3 & \mathrm{~d}_3

\end{array}\right]\) can be reduced to the form \(\left[\begin{array}{llll}

1 & 0 & 0 & \alpha \\

0 & 1 & 0 & \beta \\

0 & 0 & 1 & \gamma

\end{array}\right]\) by using elementary row transformations then x = α, y = β, z = γ, i.e., unique solution is the solution.

i) In the above matrix \(\left[\begin{array}{llll}

1 & 0 & 0 & \alpha \\

0 & 1 & 0 & \beta \\

0 & 0 & 1 & \gamma

\end{array}\right]\) is called final matrix of the system of equations.

ii) In the final matrix, if ‘O’ is obtained in place of 1 and in the same rows last element α or β or γ is

a) ‘O’ then the system has infinite number of solutions.

b) non-zero then the system of equations has no solution.

→ The system of non-homogeneous equations AX = D has

- a unique solution if rank [A] = rank [AD] = 3
- Infinitely many solutions if rank [A] = rank [AD] < 3
- No solution if rank [A] * rank [AD],

→ The system of homogeneous equations AX = O has

- The trivial solution only if Rank [A] = 3 = The number of unknowns (variables)
- An infinite number of solutions (non-trivial solution), if Rank of A less than the number of unknowns (variables) < 3.