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 = [aij] 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| = kn |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 = A3 ⇒A = 5
If A2 + A - I = 0, then A-1 is ____.
Given that A2 + A - I = 0 Premultiplying both sides by A-1, (A-1 A)A + A-1 A - A-1I = 0 ⇒ IA + I - A-1= 0 (since, A-1A = 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 AT = A ⇒ (AT)-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| = 52 |A| [matrix A is of order 2 × 2] = 25 |A| = 25 × 3 = 75