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Standard XII

Mathematics

Involutory Matrix

The trace of ...

Question

The trace of a square matrix is defined as the sum of the principal diagonal elements. For real numbers $a$ and $b,$ if the trace of matrices $A = [\begin{matrix} 2 a^{2} & 5 3 & 9 - 6 b \end{matrix}]$ and $B = [\begin{matrix} - b^{2} & 2 3 & 8 a - 8 \end{matrix}]$ are equal, then $2 a - b$ is equal to

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Solution

The correct option is B $1$
From given matrices,
$T r (A) = 2 a^{2} + 9 - 6 b$
and $T r (B) = - b^{2} + 8 a - 8$
Given, $T r (A) = T r (B)$
$\Rightarrow 2 a^{2} + 9 - 6 b = - b^{2} + 8 a - 8 \Rightarrow 2 (a^{2} - 4 a + 4) + (b^{2} - 6 b + 9) = 0 \Rightarrow 2 (a - 2)^{2} + (b - 3)^{2} = 0$
$\Rightarrow a - 2 = 0$ and $b - 3 = 0$
$\Rightarrow a = 2, b = 3$
So, $2 a - b = 1$

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Similar questions

Q. If the trace of two matrices

A = [\begin{matrix} 2 a^{2} & 5 3 & 9 - 6 b \end{matrix}]

and

B = [\begin{matrix} - b^{2} & 5 3 & 8 a - 8 \end{matrix}]

are equal,where

a, b \in R,

then

2 a - b

equals to

Q. Let

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 1 & 0 & 1 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦

satisfies

A^{n} = A^{n - 2} + A^{2} - I

for

n \geq 3.

And trace of a square matrix

X

is equal to the sum of elements in its principal diagonal. Further consider a matrix

\cup_{3 \times 3}

with its column as

\cup_{1}, \cup_{2}, \cup_{3}

such that

A^{50} \cup_{1} = ⎡ ⎢ ⎣ \begin{matrix} 12525 \end{matrix} ⎤ ⎥ ⎦, A^{50} \cup_{2} = ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦, A^{50} \cup_{3} = ⎡ ⎢ ⎣ \begin{matrix} 001 \end{matrix} ⎤ ⎥ ⎦

Then,

Trace of

A^{50}

equals

Q. If

A

is a

3 \times 3

skew symmetric matrix with real entries and trace of

A^{2}

equals zero, then

Note: Trace of matrix

A

denotes the sum of diagonal elements of matrix

A

Q. Let

S

be the set of all

3 \times 3

matrices having three entries equal to

1

and six entries equal to

0.

A matrix

M

is picked uniformly at random from the set

S .

Then which of the following statements is (are) CORRECT ?
(The trace of a square matrix is defined to be the sum of elements on the main diagonal)

Q. Let

A

and

B

are two square matrices such that the trace of

A B = 30,

then the trace of

B A

is equal to

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The trace of a square matrix is defined as the sum of the principal diagonal elements. For real numbers a and b, if the trace of matrices A=[2a2539−6b] and B=[−b2238a−8] are equal, then 2a−b is equal to

The correct option is B 1From given matrices, Tr(A)=2a2+9−6b and Tr(B)=−b2+8a−8 Given, Tr(A)=Tr(B) ⇒2a2+9−6b=−b2+8a−8⇒2(a2−4a+4)+(b2−6b+9)=0⇒2(a−2)2+(b−3)2=0 ⇒a−2=0 and b−3=0 ⇒a=2,b=3 So, 2a−b=1