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Mathematics

Find the values of a and b if,

[a+3b2+20−6]=[2a+13b0b2−5b].\begin{bmatrix}[r] a + 3 & b^2 + 2 \ 0 & -6 \end{bmatrix} = \begin{bmatrix}[r] 2a + 1 & 3b \ 0 & b^2 - 5b \end{bmatrix}.

Matrices

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Answer

By definition of equality of matrices we get,

⇒ a + 3 = 2a + 1 or a = 2

⇒ b2 + 2 = 3b     (…Eq 1)

⇒ b2 - 5b = -6     (…Eq 2)

Solving Eq 1 for b,

⇒ b2 + 2 = 3b
⇒ b2 - 3b + 2 = 0
⇒ b2 - 2b - b + 2 = 0
⇒ b(b - 2) - 1(b - 2) = 0
⇒ (b - 1)(b - 2) = 0
⇒ b = 1 or b = 2.

Checking whether the value of b = 1 satisfies Eq 2

⇒ b2 - 5b = -6

L.H.S. = b2 - 5b = (1)2 - 5(1) = -4.

L.H.S. ≠\neq R.H.S., so b = 1 is not the solution.

Checking whether the value of b = 2 satisfies Eq 2

⇒ b2 - 5b = -6

L.H.S. = b2 - 5b
= (2)2 - 5(2)
= 4 - 10
= -6 = R.H.S..

∴ a = 2 and b = 2.

Hence, the values are a = 2 and b = 2.

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