If A = [a^2 ab ac; ab b^2 bc; ac bc c^2] and a^2 + b^2 + c^2 = 1, then A^2 =

The correct answer is Option 1. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q4 – AP-EAMCET 2026

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Home Maths MatricesQuestion #282

A = [\begin{matrix} a^{2} & a b & a c \\ a b & b^{2} & b c \\ a c & b c & c^{2} \end{matrix}]

and

a^{2} + b^{2} + c^{2} = 1

, then

A^{2} =

Clue 1 of 3: Matrix Representation

Observe the structure of the given matrix $A$ . It can be represented as an outer product of two vectors.

🧠 Full Solution Path

Step 1: Recognize the structure of matrix A✦ Active

The matrix $A$ can be expressed as an outer product of a column vector and its transpose. Let $X = [\begin{matrix} a \\ b \\ c \end{matrix}]$ . Then $A = X X^{T}$ . Let's verify this by multiplying $X$ and $X^{T}$ :

A = [\begin{matrix} a \\ b \\ c \end{matrix}] [\begin{matrix} a & b & c \end{matrix}] = [\begin{matrix} a^{2} & a b & a c \\ a b & b^{2} & b c \\ a c & b c & c^{2} \end{matrix}]

This matches the given matrix $A$ in the question.

💡 Teacher's Secret Hint

Recognizing this structure (outer product) simplifies the matrix multiplication significantly, as it allows us to use scalar properties for $X^{T} X$ .

Step 2: Calculate

A^{2}

○ Expand

We need to find $A^{2}$ . Substitute the outer product representation for $A$ :

A^{2} = A \cdot A = (X X^{T}) (X X^{T})

Using the associativity of matrix multiplication, we can regroup the terms as:

A^{2} = X (X^{T} X) X^{T}

💡 Teacher's Secret Hint

Remember that matrix multiplication is associative, i.e., $(A B) C = A (B C)$ , which is crucial for simplifying expressions like $(X X^{T}) (X X^{T})$ .

Step 3: Evaluate the inner product

X^{T} X

○ Expand

Now, let's calculate the inner product $X^{T} X$ :

X^{T} X = [\begin{matrix} a & b & c \end{matrix}] [\begin{matrix} a \\ b \\ c \end{matrix}] = a \cdot a + b \cdot b + c \cdot c = a^{2} + b^{2} + c^{2}

💡 Teacher's Secret Hint

The product $X^{T} X$ for a column vector $X$ results in a scalar value, specifically the sum of the squares of its components.

Step 4: Apply the given condition and simplify

A^{2}

○ Expand

The problem states that $a^{2} + b^{2} + c^{2} = 1$ . Therefore, from Step 3:

X^{T} X = 1

Now substitute this value back into the expression for $A^{2}$ from Step 2:

A^{2} = X (X^{T} X) X^{T} = X (1) X^{T}

Since multiplying by 1 (which is a scalar in this context) does not change the product, we get:

A^{2} = X X^{T}

💡 Teacher's Secret Hint

The condition $a^{2} + b^{2} + c^{2} = 1$ is crucial here. It essentially means that the vector $X$ is a unit vector, which simplifies its inner product to 1.

Step 5: Final Result○ Expand

From Step 1, we established that $A = X X^{T}$ . Therefore, we can conclude that:

A^{2} = A

Comparing this result with the given options, option 1 is $A$ .

💡 Teacher's Secret Hint

This type of matrix, where $A^{2} = A$ , is known as an idempotent matrix. Recognizing such properties can sometimes provide shortcuts or confirm your solution.

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Maths Question 4 – AP-EAMCET 2026

Clue 1 of 3: Matrix Representation

🧠 Full Solution Path