If a matrix is a zero matrix, then a, b, c, x, y, z are connected by:

Engineering

Mathematics

Introduction to Determinants

Properties of Determinant

Question

If a matrix $|\begin{array}{lll} (x - a)^{2} (x - b)^{2} (x - c)^{2} \\ (y - a)^{2} (y - b)^{2} (y - c)^{2} \\ (z - a)^{2} (z - b)^{2} (z - c)^{2} \end{array}| = 0$ is a zero matrix, then a, b, c, x, y, z are connected by:

a + b + c = 0, x + y + z = 0

a + b + c = 0, x = y = z

a = b = c, x + y + z = 0

None of these

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$|\begin{array}{lll} (x - a)^{2} (x - b)^{2} (x - c)^{2} \\ (y - a)^{2} (y - b)^{2} (y - c)^{2} \\ (z - a)^{2} (z - b)^{2} (z - c)^{2} \end{array}| = 0$

(x – a)² = 0 ⇒ x = ±a

(x – b)² = 0 ⇒ x = ±b

(x – c)² = 0 ⇒ x = ±c

x = a = b = c

Similarly, (y – a)² = 0 ⇒ y = a

(y – b)² = 0 ⇒ y = b

(y – c)² = 0 ⇒ y = c

y = a = b = c = x

similarly,

(z – a)² = 0 ⇒ z = a

(z – b)² = 0 ⇒ z = b

(z – c)² = 0 ⇒ z = c

so. x = y = z = a = b = c

a = b = c, x + y + z = 0

option (c) is correct