Evaluate the determinant: -a^2 ab ac ab -b^2 bc ac bc -c^2

The correct answer is Option 4. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q5 – AP-EAMCET 2026

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Home Maths DeterminantsQuestion #159

| \begin{matrix} - a^{2} & a b & a c \\ a b & - b^{2} & b c \\ a c & b c & - c^{2} \end{matrix} |

Clue 1 of 3: Determinant Properties

Look for common factors in rows or columns to simplify the determinant before expansion.

🧠 Full Solution Path

Step 1: Factor out common terms from rows✦ Active

Observe that the first row ( $R_{1}$ ) has a common factor of $a$ , the second row ( $R_{2}$ ) has a common factor of $b$ , and the third row ( $R_{3}$ ) has a common factor of $c$ . We can factor these out from the determinant.

D = | \begin{matrix} - a^{2} & a b & a c \\ a b & - b^{2} & b c \\ a c & b c & - c^{2} \end{matrix} | = a b c | \begin{matrix} - a & b & c \\ a & - b & c \\ a & b & - c \end{matrix} |

💡 Teacher's Secret Hint

Remember that when you factor out a common term from a row or column, it multiplies the entire determinant.

Step 2: Factor out common terms from columns○ Expand

Now, observe the new determinant. The first column ( $C_{1}$ ) has a common factor of $a$ , the second column ( $C_{2}$ ) has a common factor of $b$ , and the third column ( $C_{3}$ ) has a common factor of $c$ . Factor these out as well.

D = a b c \cdot a b c | \begin{matrix} - 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{matrix} | = a^{2} b^{2} c^{2} | \begin{matrix} - 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{matrix} |

💡 Teacher's Secret Hint

Factoring from rows and then from columns is a powerful technique to simplify determinants, especially when variables are involved.

Step 3: Evaluate the simplified determinant○ Expand

Now, we need to evaluate the $3 \times 3$ determinant with constant entries.

| \begin{matrix} - 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{matrix} | = - 1 ((- 1) (- 1) - (1) (1)) - 1 ((1) (- 1) - (1) (1)) + 1 ((1) (1) - (- 1) (1))

= - 1 (1 - 1) - 1 (- 1 - 1) + 1 (1 - (- 1))

= - 1 (0) - 1 (- 2) + 1 (2)

= 0 + 2 + 2 = 4

💡 Teacher's Secret Hint

Be careful with signs when expanding a determinant. The cofactor expansion uses alternating signs: $+, -, +$ for the first row.

Step 4: Combine the factors○ Expand

Multiply the factored terms $a^{2} b^{2} c^{2}$ by the value of the simplified determinant, which is $4$ .

D = a^{2} b^{2} c^{2} \cdot 4 = 4 a^{2} b^{2} c^{2}

💡 Teacher's Secret Hint

Always double-check your calculations, especially the arithmetic in the final determinant expansion.

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Maths Question 5 – AP-EAMCET 2026

Clue 1 of 3: Determinant Properties

🧠 Full Solution Path