#### The elementary transformation ____________ is not possible on a matrix.

The elementary transformation C_{1} → 3C_{2}is not possible on a matrix. The multiplication of each element of the ith column by k, where k ≠ 0 is given by C_{i }→ kC_{i}. Therefore, C_{1 }→ 3C_{1} is possible but C_{1} → 3C_{2}is not possible.
C_{1 }→ 3C_{1} is possible but C_{1} → 3C_{2}

#### In order to use elementary column operations, we write A = ___________.

In order to use elementary column operations, we write A = A = AI. If A is matrix such that A^{-1}exists, then to find A^{-1}using elementary column operations, we write A = AI and apply a sequence of column operations on A = AI, till we get I = AB.

#### If A is a square matrix of order p and if there exists another square matrix B of the same order p, such that AB = BA = I, then _________________.

If A is a square matrix of order p and if there exists another square matrix B of the same order p, such that AB = BA = I, then B is called the inverse matrix of A. If A = [a_{ij}] be a square matrix of order p. If B is another square matrix of the same order and AB = BA = I, then B is called the inverse matrix of A. It is denoted by A^{-1}. A is said to be invertible. If B is the inverse of A, then A is also the inverse of B.

#### Inverse of a square matrix, if it exists, is ________________.

Inverse of a square matrix, if it exists, is unique. Let A be a square matrix of order 'k'. If possible, let B and C be two inverses of A. Since B is the inverse of A, AB = BA = I. As C is the inverse of A, AC = CA = I. Then, B = BI = B (AC) = (BA) C = IC = C => Inverse of a square matrix, if it exists, is unique.

#### If A = [a_{ij}] is a skew-symmetric matrix of order n, then aii is equal to ____.

If A [a_{ij}] is a skew-symmetric matrix, aii is equal to 0. Hence, aii is equal to 0 for all i = 1, 2 ...n.

#### The number of elementary operations on a matrix are __________.

There are 6 operations on a matrix, 3 of which are due to rows and 3 due to columns. These operations are known as elementary transformations. 1 The interchange of any two rows or columns. 2 The multiplication of the elements of any row or column by a non-zero number. 3 The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non-zero number.

#### The total number of elements in a matrix represents a prime number. The possible orders the matrix can have is _____.

The possible orders it can have is 2. As prime numbers have exactly two factors, only two orders are possible. For, example, 5 is a prime number. The orders possible are 5 × 1 and 1 × 5.

#### The interchange of any two rows is given by ___________.

If the i^{th} and j^{th} rows are interchanged, then this transformation is indicated by R_{i }↔ R_{j}.

#### If X is any m × n matrix such that XY and YX both defined, then Y is an ____.

If XY is defined, then the number of rows of Y should be n and if YX is defined, then the number of columns of matrix Y should be m. Therefore, Y should be a matrix of order n × m.

#### The inverse of a symmetric matrix is ____.

The inverse of a symmetric matrix is also a symmetric matrix.

#### The addition of constant multiplication of the elements of any row to the corresponding element of any other row is denoted by __________________.

The addition of the elements of i^{th} row with the corresponding elements of j^{th} row, multiplied by k is denoted by Ri → Ri + kR_{j}.

#### If A and B are invertible matrices of the same order, then (AB)^{-1}= _______________.

If A and B are invertible matrices of the same order, then (AB)^{-1} = B^{-1} A^{-1}. Let A and B be invertible matrices of the same order, then (AB)( AB)^{-1}= I (by definition of inverse of a matrix) Pre-multiplying by A^{-1}, A^{-1} (AB)(AB)^{-1} = A^{-1}I => (A^{-1}A) B(AB)^{-1 }= A^{-1 }(A^{-1}I = A^{-1}) => I B(AB)^{-1 }= A^{-1} => B(AB)^{-1} = A^{-1} Pre-multiplying by B^{-1} => (B^{-1}B) (AB)^{-1 }= B^{-1}A^{-1} => I (AB)^{-1} = B^{-1}A^{-1} Therefore, (AB)^{-1} = B^{-1}A^{-1}

#### Given a matrix A = [a_{ij}]; 2 ≤ i ≤ 4and 3 ≤ j ≤ 5 where a_{ij }= i + 2_{j}; the elements a_{23} and a_{36} are ______.

The elements a_{23} and a_{36} are 8 and 'not defined'. a_{23} : => i = 2 => j = 3 i + 2_{j} = 8 a_{36} : j = 6 , However, 3 ≤ j ≤ 5, which will not be satisfied. Therefore, a_{36} is not defined.

#### If A is a symmetric matrix and n ε N, then A^{n} is ____.

A^{T} = A (A is a symmetric matrix) (A^{T})^{n} = A^{n} ⇨ (A^{n})^{T} = A^{n} Hence A^{n} is a symmetric matrix.

#### If A is a symmetric matrix of integers with zeroes on the main diagonal, the sum of the entries of A must be an ____.

The sum of the entries of A must be an even integer. If sum of the entries of A is not an even integer, matrix A cannot be a symmetric matrix.

#### If A, B are symmetric matrices of the same order, then AB - BA is a ____.

A and B are symmetric matrices. Therefore, A' = A and B' = B (AB - BA)' = (AB)' - (BA)' = B'A' - A'B' = BA - AB (Since, A' = A and B' = B) = - (AB - BA) ⇨ AB - BA is skew - symmetric.

#### The order of the matrix [3 5 -7] is _____.

The order the matrix [3 5 -7] is 1 × 3 as there is only one row and 3 columns.