Matrices and Determinants Test - 1

2^3x3  = 2⁹  = 512.

The number of elements in a  3  X  3  matrix is the product  3  X  3=9.

Each element can either be a 0 or a 1.

Given this, the total possible matrices that can be selected is  2⁹ =512

If any row or column of a square matrix is 0 , then its product with itself is always a zero matrix.

The matrix given in the image is a diagonal matrix .

• A diagonal matrix is one in which all the elements outside the main diagonal are zero, and the elements on the main diagonal can be any value.
• Here, the non-zero entries (6, 3, 9) are all on the main diagonal, and all other elements are zero.

Correct answer: c) diagonal matrix .

If AB = BA = I , then A and B are inverse of each other. i.e. A is invers of B and B is inverse of A.

In  linear algebra, the  identity matrix , or sometimes ambiguously called a  unit matrix , of size  n  is the  n  × n  square matrix  with ones on the  main diagonal  and zeros elsewhere. It is denoted by  I_n , or simply by  I  if the size is immaterial or can be trivially determined by the context

The product of any matrix with itself can be found only when it is a square matrix.i.e. m = n.

If A and B are square matrices of same order , then , product of the matrices is not commutative.Therefore , the given result is true only when AB = BA.

The given system is:

Therefore, the correct answer is: λ=2.

, therefore matrix A has 4 elements in each row

Let matrix A is of order m x n , and matrix B is of order p x q . then , the product AB is defined only when n = p. that ’s why, If A and B are any two matrices, then AB may or may not be defined.

Adj.(KA) = K^{n −1}  Adj.A , where K is a scalar and A is a n x n matrix.

For No solutions,   for given system of equations we have  

Here, matrix P is of order  2  × 3 and matrix Q is of order  2  × 2 , then , the product PQ is defined only when : no. of columns in P = no. of rows in Q. And the order of resulting matrix is given by : rows in P x columns in Q.

Lower triangular matrix is given by : 
 ,

here , a_ij  = 0
if i is less than j.and a_ij  ≠ 0, if i is greater than j.

By the property of transpose , (AB)’= B ’A ’.

The orthogonal projection on x- axis is given by : 

3^2x2  = 3⁴  = 81

Upper Triangular matrix is given by  :
.
 Here, a_ij =0 , if i is greater than j.and  a_ij  ≠0, if I is less than j.

For every square matrix (A + A ’) is always symmetric.

The given system of equations does not have solution if : 

- 0  ⇒ 1(-14) - 4(-7) -2(7) = 0

For a non trivial solution : 

⇒bc + ab - 2ac = 0 ⇒  ∴there, a , b ,c, are in H.P