Test: Matrix Operations - JEE MCQ

As the given two matrices are 3x3 and 3x1 respectively
hence, the product will be a 3x1 matrix
so, {(1*0+ 2*2+ 0*x) (2*0+ 0*2+ 1*x) (1*0+ 0*2+ 2*x)}
= {4, x, 2x} as a 3x1 matrix

If A, B are, respectively m × n, k × l matrices, then both AB and BA are defined if and only if n = k and l = m. In particular, if both A and B are square matrices of the same order, then both AB and BA are defined.

A = [2, 3, 4]  
Therefore AXB = {(2*1) + (3*(-1)) + (4*2)}
AXB = {2 + (-3) + 8}
AXB = 7

A.B = [(-1(-1) + 2(-2) + 3(-3)   -1(-3) + 2(1) + 3(2)]

A.B = [1 - 4 - 9     3 + 2 + 6]

A.B = [-12   11]

 P(n) : Aⁿ = {(1+2n, -4n), (n,(1 - 2n))}
= P(k + 1) = {(1+2(k+1), -4(k+1)), (k+1, (1 - 2(k+1)}
= {(1+2k+2, -4k-4) (k+1, 1-2k-2)}
= {(2k+3, -4k-4), (k+1, -2k-1)}

Value of Determinant remains unchanged if we add equal multiples of all the elements of row (column) to corresponding elements of another row (column) If, we have a given matrix A.

A = {(2),(3)}      B = {-1,2,-2}
AB = {(-2,4,-4) (-3,6,-6)}

Determinant of a skew symmetric even ordered matrix A is a non zero perfect square.

In matrix 1*3 is one row and 3 columns and in 3*4 is three rows and four column hence multiplied matrix will be 1*4.

A={(-1,2) (3,-2) (-4,3)}     B={(1,3) (3,-2) (6,2)}
2B = {(2,6) (6,-6) (12,4)}
A - 2B = {(-1,2) (3,-2) (-4,3)} - {(2,6) (6,-6) (12,4)}
= {(-1-2, 2-6) (3-6, -2+4) (-4-12, 3-4)}
= {(-3,-4) (-3,2) (-16, -1)}

AB = 0 does not necessarily imply that either A or B is a null matrix 
- Both matrices need not be null matrices.