Let A be a 2 - 2 real matrix with entries from 0-1 and - A - - 0 - Consider the following two statements-P If A - I2- then - A - -1Q If -A-1- then trA -2-where I2 denotes 2 - 2 identity matrix and trA denotes the sum of the diagonal entries of A- Then -

P : A = [ 1 amp;1; 0 amp; 1 ]≠ I_2 & |A| ≠ 0 & |A| = 1 (false ) Q : |A| = [ 1 amp;1; 0 amp; 1 ] = 1 then Tr(A ) = 2 (true ) ...

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Byju's Answer

Standard XII

Mathematics

Derivative of Some Standard Functions

Let A be a 2 ...

Question

Let $A$ be a $2 \times 2$ real matrix with entries from ${0, 1}$ and $| A | \neq 0$ . Consider the following two statements:

(P) If $A \neq I_{2},$ then $| A | = - 1$

(Q) If $| A | = 1$ , then $tr (A) = 2$ ,
where $I_{2}$ denotes $2 \times 2$ identity matrix and $tr (A)$ denotes the sum of the diagonal entries of $A$ . Then :

Both (P) and (Q) are false

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(P) is true and (Q) is false

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Both (P) and (Q) are true

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(P) is false and (Q) is true

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Solution

The correct option is D (P) is false and (Q) is true
$P : A = [\begin{matrix} 1 & 1 0 & 1 \end{matrix}] \neq I_{2} & | A | \neq 0 & | A | = 1 (false)$

$Q : | A | =$ $[\begin{matrix} 1 & 1 0 & 1 \end{matrix}] = 1$

then Tr $(A) = 2 (true)$

Suggest Corrections

Similar questions

Q. Let A be a

2 \times 2

matrix with real entries. Let I be the

2 \times 2

identity matrix. Denote by tr(A), the sum of diagonal entries of

A

. Assume that

A^{2} =

I.
Statement-l: If

A \neq I

and

A \neq - I

, then

det A = - 1

.
Statement-2: If

A \neq I

and

A \neq - I

, then

t_{1} {A) \neq 0

Q. Assertion :Let

A

be a

2 \times 2

matrix with real entries. Let

I

be the

2 \times 2

identity matrix. Denote by

t r (A)

, the sum of diagonal entries of

A

. Assume that

A^{2} = I

A \neq I

and

A \neq - I

, then

d e t (A) = - 1

Reason: If

A \neq I

and

A \neq - I

, then

t r (A) \neq 0

Q. Let

A

be a

2 \times 2

matrix with real entries, Let I be the

2 \times 2

identity matrix. Denote by

t r (A),

the sum of diagonal entries of

A

. Assume that

A^{2} = I

Statement

1 :

A \neq I

and

A \neq - I

, then det

A = - 1

Statement

2 :

A \neq I

and

A \neq - I

, then

t r (A) \neq 0,

then which of the following is correct

Q. Let A be a

2 \times 2

matrix with non-zero entries and let

A^{2} = I

, where I is

2 \times 2

identity matrix. Define Tr(A)

=

sum of diagonal elements of A and

| A | =

determinant of matrix A.
Statement-1 Tr(A)

= 0

Statement-2:

| A | = 1

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

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Let A be a 2×2 real matrix with entries from {0,1} and |A|≠0 . Consider the following two statements: (P) If A≠I2, then |A|=−1 (Q) If |A|=1, then tr(A)=2, where I2 denotes 2×2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then :

The correct option is D (P) is false and (Q) is trueP:A=[1101]≠I2&|A|≠0&|A|=1(false) Q:|A|= [1101]=1 then Tr(A)=2(true)

The correct option is D (P) is false and (Q) is true
$P : A = [\begin{matrix} 1 & 1 0 & 1 \end{matrix}] \neq I_{2} & | A | \neq 0 & | A | = 1 (false)$

$Q : | A | =$ $[\begin{matrix} 1 & 1 0 & 1 \end{matrix}] = 1$

then Tr $(A) = 2 (true)$