The number of matrices A in Λ for which the system of linear equations A {bmatrix} x \\ y \\ x {bmatrix} = {bmatrix} 1 \\ 0 \\ 0 {bmatrix} has a unique solution, is [JEE Adv 2009 P1]

Correct Answer is : at least 4 but less than 7.

Correct Answer is : at least 4 but less than 7.

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Matrices and Determinants

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Let

\Lambda

be the set of all

3 \times 3

symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0 .

Section: Mathematics

Question : 55 of 56

Marks: +1, -0

\begin{bmatrix} x \\ y \\ x \end{bmatrix}

\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}

Solution:

We have

\begin{bmatrix} 0 & a & b \\ a & 0 & c \\ b & c & 1 \end{bmatrix}

We can see that either b = 0 or c = 0 ⇒ |A| ≠ 0; therefore, two matrices.

\begin{bmatrix} 0 & a & b \\ a & 1 & c \\ b & c & 0 \end{bmatrix}

Now, either a = 0 or c = 0 ⇒ |A| ≠ 0; therefore, two matrices

\begin{bmatrix} 1 & a & b \\ a & 0 & c \\ b & c & 0 \end{bmatrix}

Now, either a = 0 or b = 0 ⇒ |A| ≠ 0; therefore, two matrices.

\begin{bmatrix} 1 & a & b \\ a & 1 & c \\ b & c & 1 \end{bmatrix}

When a = b = 0 ⇒ A = 0.

When a = c = 0 ⇒ A = 0.

When b = c = 0 ⇒ A = 0.

Therefore, there are only six matrices.

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