Let A and B are two matrices of same order 3- 3 given by A- 1 3 -2- 2 4 6- 3 5 8 - B- - 3 2 4- 3 2 5- 2 1 4 -If - -4-then 1-6trAB-trBA is equal to

The correct option is B 37 A=[ 1 amp; 3 amp; 6; 2 amp; 4 amp; 6; 3 amp; 5 amp; 8 ],B=[ 3 amp; 2 amp; 4; 3 amp; 2 amp; 5; ...

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Byju's Answer

Standard XII

Mathematics

Singular and Non Singualar Matrices

Let A and ...

Question

Let $A$ and $B$ are two matrices of same order $3 \times 3$ given by $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & λ + 2 2 & 4 & 6 3 & 5 & 8 \end{matrix} ⎤ ⎥ ⎦$ $B = ⎡ ⎢ ⎣ \begin{matrix} 3 & 2 & 4 3 & 2 & 5 2 & 1 & 4 \end{matrix} ⎤ ⎥ ⎦$ If $λ = 4$ ,then $\frac{1}{6} {t r (A B) + t r (B A)}$ is equal to

42

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37

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35

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None of these

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Solution

The correct option is B $37$
$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & 6 2 & 4 & 6 3 & 5 & 8 \end{matrix} ⎤ ⎥ ⎦, B = ⎡ ⎢ ⎣ \begin{matrix} 3 & 2 & 4 3 & 2 & 5 2 & 1 & 4 \end{matrix} ⎤ ⎥ ⎦$
$A B = ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & 6 2 & 4 & 6 3 & 5 & 8 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 3 & 2 & 4 3 & 2 & 5 2 & 1 & 4 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 24 & 14 & 43 30 & 18 & 52 40 & 24 & 69 \end{matrix} ⎤ ⎥ ⎦$

$\Rightarrow t r (A B) = 24 + 18 + 69 = 111$

$t r (B A) = t r (A B) = 111$

$\frac{1}{6} {t r (A B) + t r (B A)} = \frac{1}{6} 222 = 37$

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Similar questions

Let

A

and

B

are two matrices of same order

3 \times 3

given by

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & λ + 2 2 & 4 & 6 3 & 5 & 8 \end{matrix} ⎤ ⎥ ⎦

B = ⎡ ⎢ ⎣ \begin{matrix} 3 & 2 & 4 3 & 2 & 5 2 & 1 & 4 \end{matrix} ⎤ ⎥ ⎦

2 A + B

is singular, then

2 λ

equals

Let

A

and

B

are two matrices of same order

3 \times 3

given by

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & λ + 2 2 & 4 & 6 3 & 5 & 8 \end{matrix} ⎤ ⎥ ⎦

B = ⎡ ⎢ ⎣ \begin{matrix} 3 & 2 & 4 3 & 2 & 5 2 & 1 & 4 \end{matrix} ⎤ ⎥ ⎦

t r (A B)^{t} + t r (B A)^{t} = t r (A B)

then the value of

2 λ

equals

Let

A

and

B

are two matrices of same order

3 \times 3

given by

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & λ + 2 2 & 4 & 6 3 & 5 & 8 \end{matrix} ⎤ ⎥ ⎦,

B = ⎡ ⎢ ⎣ \begin{matrix} 3 & 2 & 4 3 & 2 & 5 2 & 1 & 4 \end{matrix} ⎤ ⎥ ⎦

A

is a singular matrix, then

t r (A + B)

is equal to

Q. Consider the following statements:
1. If

A

and

B

are two square matrices of same order, then

(A + B) (A - B) = A^{2} - B^{2}

.
2. If

A

and

B

are two square matrices of same order, then

(A B)^{n} = A^{n} B^{n}

.
3. If

A

and

B

are two matrices such that

A B = A

and

B A = B

, then

A

and

B

are idempotent.
Which of these is/are not correct?

A

and

B

are two matrices of same order

3 \times 3

, where

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 2 & 3 & 4 5 & 6 & 8 \end{matrix} ⎤ ⎥ ⎦

and

B = ⎡ ⎢ ⎣ \begin{matrix} 3 & 2 & 5 2 & 3 & 8 7 & 2 & 9 \end{matrix} ⎤ ⎥ ⎦

The value of

Adj (Adj A)

is equal to

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Let A and B are two matrices of same order 3×3 given by A=⎡⎢⎣13λ+2246358⎤⎥⎦ B=⎡⎢⎣324325214⎤⎥⎦If λ=4,then 16{tr(AB)+tr(BA)} is equal to

The correct option is B 37A=⎡⎢⎣136246358⎤⎥⎦,B=⎡⎢⎣324325214⎤⎥⎦AB=⎡⎢⎣136246358⎤⎥⎦⎡⎢⎣324325214⎤⎥⎦=⎡⎢⎣241443301852402469⎤⎥⎦⇒tr(AB)=24+18+69=111tr(BA)=tr(AB)=111 16{tr(AB)+tr(BA)}=16222=37