#### A homogeneous system of n linear equations in n unknowns is expressible in the form AX = 0. The system has non-trivial solution if ____.

A homogeneous system of n linear equations in n unknowns is expressible in the form AX = 0. A homogeneous system of equation is always consistent. If |A| ≠ 0, then AX = 0 has unique solution. This solution is called the trivial solution, where x = y = z = 0. If |A| = 0, then AX = 0 has infinitely many solutions.

#### If A = [a_{ij}] is a square matrix of order n × n and k is a scalar then |kA| is equal to ____.

If A = kB where A and B are square matrices of order n, then |A| = k^{n} |B| where n = 1, 2, 3.

#### If A and B are non singular square matrices of the same order, then the relationship between adj AB, adj A and adj B is ____.

If A and B are square matrices of the same order n, then adj (AB) = (adj B) (adj A) We know that (AB) adj (AB) = |AB|I = (adj A AB) (AB) ----- (1) (AB) (adj B. adj A) = A.B adj B. adj A = A (B adj B) adj A = A(|B|I) adj A [because B adj B = |B|I] = |B| (A adj A) = |B| |A| I [because A adj A = |A|I] = |A| |B|I = |AB| I ----- (2) From (1) and (2) AB (adj. AB) = AB (adj. B adj. A) Pre-mul. both sides by (AB)^{-1} (AB)^{-1} ((AB) adj. AB) = (AB)^{-1} ((AB) adj. B adj. A) ⇒ adj. AB = adj B. adj A

#### If A is a square matrix of order 4 such that |adj A| = 125, then |A| is ____.

We know that |adj A| = |A|n-1, where n is the order of the matrix. Therefore 125 = |A|^{4 – 1} ⇒125 = A^{3} ⇒A = 5

#### If A^{2} + A - I = 0, then A^{-1 }is ____.

Given that A^{2} + A - I = 0 Premultiplying both sides by A^{-1}, (A^{-1} A)A + A-^{1} A - A^{-1}I = 0 ⇒ IA + I - A^{-1}= 0 (since, A^{-1}A = I) ⇒ A + I - A^{-1}= 0 [IA = A] ⇒ A^{-1}= A + I

#### A square matrix A of order n is invertible if there exists a square matrix B of the same order such that AB = In = BA. In such a case, we say that the inverse of matrix A is B. Following are some properties of inverse of a matrix. The property that is not true is ____.

If A is an invertible symmetric matrix, then A^{T } = A ⇒ (A^{T})^{-1} = A^{-1} ⇒ (A^{-1})T = A^{-1} [since, (AT)^{-1} = (A^{-1})T] Therefore, the inverse of an invertible symmetric matrix is a symmetric matrix. Hence, the statement given in Option A is false.

#### If for matrix A, |A| = 3, where matrix A is of order 2 × 2, then |5 A| is ____.

|A| = 3 (given) |5 A| = 5^{2} |A| [matrix A is of order 2 × 2] = 25 |A| = 25 × 3 = 75