Using atleast three properties of determinants show that the determinant of the 3×3 matrix with rows [a²+1, ab, ac], [ab, b²+1, bc], [ca, cb, c²+1] equals 1 + a² + b² + c².

**OR**

Using atleast three properties of determinants, solve: determinant of the 3×3 matrix with rows [a+

Question

Using atleast three properties of determinants show that the determinant of the 3×3 matrix with rows [a²+1, ab, ac], [ab, b²+1, bc], [ca, cb, c²+1] equals 1 + a² + b² + c².

**OR**

Using atleast three properties of determinants, solve: determinant of the 3×3 matrix with rows [a+x, a−x, a−x], [a−x, a+x, a−x], [a−x, a−x, a+x] = 0.

Accepted Answer

>

> [!TLDR]
> Multiplying rows by $a,b,c$ and factoring columns reduces the determinant to a simple triangular form giving $1+a^2+b^2+c^2$.
>
> Let $D = \begin{vmatrix} a^2+1 & ab & ac \ ab & b^2+1 & bc \ ca & cb & c^2+1 \end{vmatrix}$
>
> **Step 1:** Multiply $R_1$ by $a$, $R_2$ by $b$, $R_3$ by $c$ (this multiplies $D$ by $abc$), then factor $a$ from $C_1$, $b$ from $C_2$, $c$ from $C_3$ (this divides back by $abc$). Net effect:
>
> $D = \begin{vmatrix} a^2+1 & a^2 & a^2 \ b^2 & b^2+1 & b^2 \ c^2 & c^2 & c^2+1 \end{vmatrix}$
>
> **Step 2:** Apply $C_1	o C_1-C_3$ and $C_2	o C_2-C_3$:
>
> $D = \begin{vmatrix} 1 & 0 & a^2 \ 0 & 1 & b^2 \ -1 & -1 & c^2+1 \end{vmatrix}$
>
> **Step 3:** Apply $R_3 	o R_3+R_1+R_2$:
> …

Concept understanding — Determinant Evaluation Using Identities

Determinant Evaluation Using Identities

The geometric intuition

The three row (or column) operations

The strategy

Worked example

Why determinants change under row operations

Step-by-step evaluation

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