Use these Inter 1st Year Maths 1A Formulas PDF Chapter 3 Matrices to solve questions creatively.

## Intermediate 1st Year Maths 1A Matrices Formulas

→ An ordered rectangular array of elements is called a matrix.

→ A matrix in which the number of rows is equal to the number of columns, is called a square matrix. Otherwise it is called rectangular matrix. In a square matrix, an element a_{ij} is in principal diagonal, if i = j

→ If each non-diagonal element of a square matrix is equal to zero, then it is called a diagonal matrix.

→ If each non-diagonal element of a square matrix is equal to zero and each diagonal element is equal to a scalar, then it is called a scalar matrix.

→ If each non-diagonal element of a square matrix is equal to zero and each diagonal element is equal to 1, then that matrix is called a unit matrix or Identity matrix.

→ If A is a square matrix then the sum of elements in the principal diagonal of A is called trace of A.

- Matrix addition is commutative.
- Matrix addition is associative.
- Matrix multiplication is associative.
- Matrix multiplication is distributive over matrix addition.

→ A square matrix A is said to be an idempotent matrix if A^{2} = A.

→ A square matrix A is said to be an involuntary matrix if A^{2} = I.

→ A square matrix A is said to be a nilpotent matrix, if there exist a positive integer n such that A^{n} = 0. If n is the least positive integer such thatAn = 0, then n is called index of the nilpotent matrix A.

→ The matrix obtained by interchanging rows and columns is called transpose of the given matrix. Transpose of A is denoted by A^{T} (or) A’.

→ For any matrices A and B

- (A
^{T})^{T}= A - (A + B)
^{T}= A^{T}+ B^{T}(A and B are same order) - (AB)
^{T}= B^{T}A^{T}. (If A and B are of orders m xn and n xp respectively)

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

→ A square matrix A is said to be skew symmetric if A^{T} = A.

→ Every square matrix can be uniquely expressed as a sum of symmetric matrix and a skew symmetric matrix.

→ The minor of an element in the square matrix of order ‘3’ is defined as the determinant of 2 × 2 matrix, obtained after deleting the row and the column in which the element is present.

→ The cofactor of an element in the i^{th} row and j^{th} column of 3 × 3 matrix is defined as its minor multiplied by (-1)^{i+j}.

→ The sum of the products of the elements of any row or column with their corresponding cofactors is called the determinant of a matrix.

- A square ‘A’ is said to be a singular matrix if det A = 0.
- A square A’ is said to be a non-singular matrix if det A ≠ 0.

→ 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 and it is denoted by adj (A).

- A square 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 and it is denoted by A
^{-1}. - If A is an invertible then A
^{-1}= 1.

→ A square matrix A is non-singular if A is invertible.

- If A and B are non-singular matrices of same type then Adj (AB) = (Adj B) (Adj A).
- If A is a square matrix of type n then det (Adj A) = (det A)
^{n-1}.

→ A matrix obtained by deleting some rows or columns (or both) of a matrix is called a submatrix.

→ Let Abe a non-zero matrix, the rank of A is defined as the maximum of the orders of the non-singular square submatrices of A. The rank of a null matrix is zero. The rank of a matrix A is denoted as rank (A) or P (A).

→ A system of linear equations is

- Consistent, if it has a solution
- Inconsistent, if it has no solution

→ Non homogeneous system

- 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}

→ The above system of equations has

- aunique solution if rank (A) = Rank [AD] = 3
- infinity many solutions, if rank (A) = Rank ([AD]) < 3
- no solution, if rank A ≠ Rank ([AD])

→ Homogeneous system of equations

- 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}

→ The above system has

- Trival solution x = y = z = 0 only if rank (A) = 3
- infinitely many non-trival solutions if rank (A) < 3.

→ A matrix is an arrangement of real or complex numbers into rows and columns so that all the rows (columns) contain equal no. of elements.

→ If a matrix consists of ‘m’ rows and ‘n’ columns, then it is said to be of order m × n.

→ A matrix of order n × n is said to be a square matrix of order n.

→ A matrix (a_{ij})_{m×n} is said to be a null matrix if a_{ij} = 0 for all i and j.

→ Two matrices of the same order are said to be equal if the corresponding elements in the matrices are all equal.

→ A matrix (a_{ij})_{n×n} is a diagonal matrix a_{ij} = 0 for all i ≠ j

→ A matrix (a_{ij})_{n×n} is a scalar matrix if a = 0 for all i ≠ j and a_{ij} = k (constant) for i = j

→ A matrix (a_{ij})_{n×n} is said to be a unit matrix of order n, denoted by I_{n} if a_{ij} = 1, when i = j and a_{ij} = 0 when i ≠ j

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]\)

→ If A = (a_{ij})_{m×n}, B = (b_{ij})_{m×n}, then A + B = (a_{ij} + b_{ij})_{m×n}

→ Matrix addition is commutative and associative

→ Matrix multiplication is not commutative but associative

→ If A is a matrix of order m × n, then AI_{n} = I_{m}A = A(AI = IA = A)

→ If AB = CA = I, then B = C

→ If A = (a_{ij})_{m×n}, then A ^{T} = (a_{ij})_{n×m}

→ (KA)^{T} = KA^{T}, (A + B)^{T} = A^{T} + B^{T}, (AB)^{T} = B^{T}.A^{T}

→ A(B + C) = AB + AC, (A + B)C = AC + BC

→ A square matrix is said to be “non-singular” if detA ≠ 0

→ A square matrix is said to be “singular” if detA = 0

→ If AB = 0, where A and B are non-zero square matrices, then both A are singular.

→ A minor of any element in a square matrix is determinant of the matrix obtained by omitting the row and column in which the element is present.

→ In (a_{ij})_{n×n}, the cofactor of a_{ij} is (-1)^{i+j} × (minor of a_{ij}).

→ In a square matrix, the sum of the products of the elements of any row (column) and the corresponding cofactors is equal to the determinant of the matrix.

→ In a square matrix, the sum of the products of the elements of any row (column) and the corresponding cofactors of any other row (column) is alway s zero.

→ If A is any square matrix, then A adjA = adjA. A = detA. I

→ If A is any square matrix and there exists a matrix B such that AB = BA = I, then B is called the inverse of A and denoted by A^{-1}.

→ AA^{-1} = A^{-1}A = I.

→ If A is non-singular, then A^{-1} = \(\frac{{adj} A}{{det} A}\) (or) adj A = |A|AA^{-1}

→ If A = \(\left(\begin{array}{ll}

a & b \\

c & d

\end{array}\right)\), then A^{-1} = \(\frac{1}{a d-b c}\left(\begin{array}{cc}

d & -b \\

-c & a

\end{array}\right)\)

→ (A^{-1})^{-1} = A, (AB)^{-1} = B^{-1}.A^{-1}, (A^{-1})^{T} =( A^{T})^{-1}; (ABC….)^{-1} = C^{-1}B^{-1}A^{-1}.

Theorem:

Matrix multiplication is associative. i.e. if conformability is assured for the matrices A, B and C, then (AB)C = A(BC).

Proof:

Theorem:

Matrix multiplication is distributive over matrix addition i.e. if conformability is assured for the matrices A, B and C, then

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

(ii) (B + C) A = BA + CA

Proof:

Let A = (a_{ij})_{m×n}, B = (b_{jk})_{n×p} C = (c_{ki})_{n×p}

B + C = (d_{jk})_{n×p}, where d_{jk} = b_{jk} + c_{jk}

∴ A(B + C) = AB + AC

Similarly we can prove that

(B + C) = BA + CA.

Theorem:

If A is any matrix, then (A^{T})^{T} = A.

Proof:

Let A = (a_{ij})_{m×n}

A^{T} = (a’_{jk})_{n×m}, where a’_{ji} = a_{ij}

(A^{T})^{T} = (a”_{ji})_{m×n}, where a”_{ij} = a_{ji}

a”_{ij} = a’_{ji} = a_{ij}

∴ (A^{T})^{T} = A

Theorem:

If A and B are two matrices o same type, then (A + B)^{T} = A^{T} + B^{T}.

Proof:

Let A = (a_{ij})_{m×n}, B = (b_{ij})

A + B = (c_{ij})_{m×n},where c_{ij} = a_{ij} + b_{ij}

(A + B)^{T} = (c’_{ji})_{n×m}. c’_{ji} = c_{ij}

A^{T} = (a’_{ji})_{n×m},where a’_{ji} = a_{ij}

B^{T} = (b_{ji})_{n×m}. where. b’_{kj} = b_{jk}

A^{T} + B^{T} = (d_{ji})_{n×m}, where d_{ji} = a’_{ji} + b’_{ji}

c’_{ji} = c_{ij} = a_{ij} + b_{ij} = a’_{ji} + b’_{ji} = d_{ji}

∴(A + B)^{T} = A^{T} + B^{T}

Theorem:

If A and B are two matrices for which conformability for multiplication is assured, then (AB)^{T} = B^{T}A^{T}.

Pr0of:

Let A = (a_{ij})_{m×n}, B = (b_{ji})_{n×p}

AB = (c_{ik})_{m×p}, where c_{ik} = \(\sum_{j=1}^{n}\) a_{ij}b_{jk}

(AB)^{T} = (c_{ki})_{p×m},where c_{ki} = c_{ik}

A^{T} = (a_{ji})_{n×m},where a_{ji} = a_{ij}

B^{T} = (b_{kj})_{p×n}, where b_{kj} = b_{jk}

B^{T} . A^{T} = (d_{ki} )_{p×m}, where d_{ki}= \(\sum_{j=1}^{n}\) b_{kj}a_{ji}

c’ki = cik = \(\sum_{j=1}^{n}\) a_{ij}b_{jk}= \(\sum_{j=1}^{n}\) b_{kj}a_{ji}d_{ki}

∴ (AB)^{T} = B^{T}A^{T}

Theorem:

If A and B are two invertible matrices of same type then AB is also invertible and (AB)^{-1} = B^{-1}A^{-1}.

Proof:

A is invertible matrix ⇒ A^{-1} exists and AA^{-1} = A^{-1}A = I.

B is an invertible matrix ⇒ B^{-1} exists and

BB^{-1} = B^{-1}B = I

Now (AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = AIA^{-1} = AA^{-1} = I

(B^{-1}A^{-1})(AB) = B^{-1}(A^{-1}A)B ∴ AB is invertible and

= B^{-1}IB = B^{-1}B = I

(AB)(B^{-1}A^{-1}) = (B^{-1}A^{-1}) = (B^{-1}A^{-1})(AB) = 1

(AB)^{-1} = B^{-1}A^{-1}.

Theorem:

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

Proof:

Let A = \(\left[\begin{array}{lll}

\mathrm{a}_{1} & \mathrm{~b}_{1} & \mathrm{c}_{1} \\

\mathrm{a}_{2} & \mathrm{~b}_{2} & \mathrm{c}_{2} \\

\mathrm{a}_{3} & \mathrm{~b}_{3} & \mathrm{c}_{3}

\end{array}\right]\) be a non – singular matrix.

∴ det A ≠ 0