JEE Mathematics 2023 Question with Solution

Given: A square matrix $A$ has order $m$ and $\det(A)=m-n$. Also,

and

Find: The value of $a+b+c$ if

First, find $m$ and $n$. Multiply

by $4$:

Subtracting this from

gives

so

Substituting into $4m+n=22$:

therefore

Now the order of the matrix is $5$, and

Use the determinant identities

and

Evaluate $\det(mA)$:

Hence,

Now evaluate the determinant of the adjoint again:

Multiply by $n$ inside the determinant. Since the matrix order is $5$,

Thus,

Express this as $3^{a}5^{b}6^{c}$. Since

compare powers:

Therefore,

So, the correct option is B.

Given: $m$ and $n$ satisfy two linear equations, and the matrix order is $m$. Find: $a+b+c$ from the prime-power form of the determinant.

After solving the equations, get $m=5$ and $n=2$, so $\det(A)=3$. Then directly chain the determinant rules:

Finally, multiplying the matrix by $n=2$ contributes a factor of $2^{5}$ to the determinant, so

Now match

So immediately,

Hence,

This works because $6^{c}$ packages one factor of $2$ and one factor of $3$ together.