Explanation

Orthogonal matrix definition: Aᵀ = A⁻¹  

Multiply both sides by A → AᵀA = I

Last verified on 05-06-2026

Explanation

Sum of coefficients = put a = 1, b = 1 in expression  

(a - b)¹⁰ → (1 - 1)¹⁰ = 0¹⁰ = 0

Last verified on 05-06-2026

Explanation

u = x² + y²  

First take ∂u/∂y = 0 + 2y = 2y  

Now take ∂/∂x of 2y = 0, because 2y has no x

So ∂²u/∂x∂y = 0

Last verified on 05-06-2026

Explanation

RHS all 0 → homogeneous system  

Find det of 3 2 1; 2 1 4; 1 4 1 = -34  

det ≠ 0 → only x1=x2=x3=0 works

det = 0 would mean infinite solutions. Here it’s not 0.

Last verified on 05-06-2026

Explanation

Parabola: y = x² - 4  

Circle: x² + y² = 9  

Put y: x² + (x² - 4)² = 9  

x⁴ - 7x² + 7 = 0  

Let t = x²: t² - 7t + 7 = 0  

D = 49 - 28 = 21 > 0  

Both t roots positive => 2 x values each  

Total intersections = 4

Last verified on 05-06-2026