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

Matrices and Determinants PYQ — JEE Main 2020 · 2 Sep · Shift 1 · Q25 with Solution

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Correct answer: D

Let $A=\begin{pmatrix}a&b\\ c&d\end{pmatrix},\quad a,b,c,d\in\{0,1\}$ with $|A|\ne 0.$ Also, $|A|=ad-bc.$ Since all entries are $0$ or $1$ , the determinant can only be in $\{-1,0,1\}$ .

We analyze all invertible $2\times 2$ matrices with entries from $\{0,1\}$ .

1. Find all such matrices with non-zero determinant

We need $ad-bc\ne 0.$ So $ad-bc=\pm 1$ .

Checking possibilities, the invertible matrices are:

$\begin{pmatrix}1&0\\0&1\end{pmatrix}=I_2, \quad |A|=1,\quad \operatorname{tr}(A)=2$
$\begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad |A|=-1,\quad \operatorname{tr}(A)=0$
$\begin{pmatrix}1&1\\0&1\end{pmatrix}, \quad |A|=1,\quad \operatorname{tr}(A)=2$
$\begin{pmatrix}1&0\\1&1\end{pmatrix}, \quad |A|=1,\quad \operatorname{tr}(A)=2$
$\begin{pmatrix}1&1\\1&0\end{pmatrix}, \quad |A|=-1,\quad \operatorname{tr}(A)=1$
$\begin{pmatrix}0&1\\1&1\end{pmatrix}, \quad |A|=-1,\quad \operatorname{tr}(A)=1$

So there are exactly $6$ invertible matrices.

2. Check statement (P)

Statement (P): If $A\ne I_2$ , then $|A|=-1$ .

This is false because there are matrices different from $I_2$ with determinant $1$ , for example $A=\begin{pmatrix}1&1\\0&1\end{pmatrix}\ne I_2$ but $|A|=1.$

Hence, (P) is false.

3. Check statement (Q)

Statement (Q): If $|A|=1$ , then $\operatorname{tr}(A)=2$ .

From the list above, matrices with determinant $1$ are:

\begin{pmatrix}1&1\\0&1\end{pmatrix}, \begin{pmatrix}1&0\\1&1\end{pmatrix}$$ Each of these has trace $$1+1=2.$$ So **(Q) is true**. We can also justify this directly: If $|A|=1$, then $$ad-bc=1.$$ Since $a,b,c,d\in\{0,1\}$, we must have $ad=1$ and $bc=0$. Thus $$a=d=1,$$ and hence $$\operatorname{tr}(A)=a+d=2.$$ So (Q) is indeed true. --- ## 4. Final conclusion - (P) is false - (Q) is true Therefore, the correct option is $$\boxed{\text{D}}$$

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1. Find all such matrices with non-zero determinant

We need

ad-bc\ne 0.

ad-bc=\pm 1

Checking possibilities, the invertible matrices are:

\begin{pmatrix}1&0\\0&1\end{pmatrix}=I_2, \quad |A|=1,\quad \operatorname{tr}(A)=2

\begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad |A|=-1,\quad \operatorname{tr}(A)=0

\begin{pmatrix}1&1\\0&1\end{pmatrix}, \quad |A|=1,\quad \operatorname{tr}(A)=2

\begin{pmatrix}1&0\\1&1\end{pmatrix}, \quad |A|=1,\quad \operatorname{tr}(A)=2

\begin{pmatrix}1&1\\1&0\end{pmatrix}, \quad |A|=-1,\quad \operatorname{tr}(A)=1

\begin{pmatrix}0&1\\1&1\end{pmatrix}, \quad |A|=-1,\quad \operatorname{tr}(A)=1

So there are exactly

6

invertible matrices.

2. Check statement (P)

Statement (P): If

A\ne I_2

, then

|A|=-1

This is false because there are matrices different from

I_2

with determinant

1

, for example

A=\begin{pmatrix}1&1\\0&1\end{pmatrix}\ne I_2

but

|A|=1.

Hence, (P) is false.

3. Check statement (Q)

Statement (Q): If

|A|=1

, then

\operatorname{tr}(A)=2

From the list above, matrices with determinant

1

are:

\begin{pmatrix}1&1\\0&1\end{pmatrix}, \begin{pmatrix}1&0\\1&1\end{pmatrix}$$ Each of these has trace $$1+1=2.$$ So **(Q) is true**. We can also justify this directly: If $|A|=1$, then $$ad-bc=1.$$ Since $a,b,c,d\in\{0,1\}$, we must have $ad=1$ and $bc=0$. Thus $$a=d=1,$$ and hence $$\operatorname{tr}(A)=a+d=2.$$ So (Q) is indeed true. --- ## 4. Final conclusion - (P) is false - (Q) is true Therefore, the correct option is $$\boxed{\text{D}}$$

Matrices and Determinants question

Matrices and Determinants question

1. Find all such matrices with non-zero determinant

2. Check statement (P)

3. Check statement (Q)

More from Matrices and Determinants

Matrices and Determinants question

Matrices and Determinants question

1. Find all such matrices with non-zero determinant

2. Check statement (P)

3. Check statement (Q)

More from Matrices and Determinants