if A= beginpmatrix2&-1 -7&4 endpmatrix and B= beginpmatrix4&1 7&2 endpmatrix then which statement is true ?

Question

if A= beginpmatrix2&-1 -7&4 endpmatrix and B= beginpmatrix4&1 7&2 endpmatrix then which statement is true ? (A) AAT = I (B) BBT = I (C) AB ≠ BA (D) (AB)T = I

Tardigrade Admin · Accepted Answer

Here AAT= beginpmatrix2&-1 -7&4 endpmatrix beginpmatrix2&-7 -1&4 endpmatrix≠ beginpmatrix1&0 0&1 endpmatrix (BBT)11=(d)2+(a)2≠1 ∴ AB≠ BA may be not true Now AB beginpmatrix2&-1 -7&4 endpmatrix beginpmatrix4&1 7&2 endpmatrix = beginpmatrix8-7&2-2 -28+28&-7+8 endpmatrix= beginpmatrix1&0 0&1 endpmatrix ; (AB)T= beginpmatrix1&0 0&1 endpmatrix=I

Q. if $A = (2 - 7 - 1 4)$ and $B = (4712)$ then which statement is true ?

$A A^{T} = I$

$B B^{T} = I$

$A B \neq = B A$

$(A B)^{T} = I$

Here $A A^{T} = (2 - 7 - 1 4) (2 - 1 - 7 4) \neq = (1001)$
$(B B^{T})_{11} = (d)^{2} + (a)^{2} \neq = 1$
$∴ A B \neq = B A$ may be not true
Now $A B (2 - 7 - 1 4) (4712)$
$= (8 - 7 - 28 + 28 2 - 2 - 7 + 8) = (1001); (A B)^{T} = (1001) = I$

Q. if A=(2−7​−14​) and B=(47​12​) then which statement is true ?

AAT=I

BBT=I

AB=BA

(AB)T=I