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Mathematics

Given matrix B = [1183]\begin{bmatrix} 1 & 1 \ 8 & 3 \end{bmatrix}. Find the matrix X if, X = B2 - 4B. Hence, solve for a and b given X × [ab]\begin{bmatrix} a \ b \end{bmatrix} = [550]\begin{bmatrix} 5 \ 50 \end{bmatrix}.

Matrices

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Answer

Given,

B = [1183]\begin{bmatrix} 1 & 1 \ 8 & 3 \end{bmatrix}

X = B2 - 4B

Solving for B2:

⇒B2=[1183]×[1183]=[(1)(1)+(1)(8)(1)(1)+(1)(3)(8)(1)+(3)(8)(8)(1)+(3)(3)]=[1+81+38+248+9]=[943217].\Rightarrow B^2 = \begin{bmatrix} 1 & 1 \ 8 & 3 \end{bmatrix} \times \begin{bmatrix} 1 & 1 \ 8 & 3 \end{bmatrix} \\[1em] = \begin{bmatrix} (1)(1) + (1)(8) & (1)(1) + (1)(3) \ (8)(1) + (3)(8) & (8)(1) + (3)(3) \end{bmatrix} \\[1em] = \begin{bmatrix} 1 + 8 & 1 + 3 \ 8 + 24 & 8 + 9 \end{bmatrix} \\[1em] = \begin{bmatrix} 9 & 4 \ 32 & 17 \end{bmatrix}.

Solving for 4B:

⇒4B=4[1183]=[443212].\Rightarrow 4B = 4\begin{bmatrix} 1 & 1 \ 8 & 3 \end{bmatrix} \\[1em] = \begin{bmatrix} 4 & 4 \ 32 & 12 \end{bmatrix}.

Now X = B2 - 4B:

⇒X=[943217]−[443212]=[9−44−432−3217−12]=[5005].\Rightarrow X = \begin{bmatrix} 9 & 4 \ 32 & 17 \end{bmatrix} - \begin{bmatrix} 4 & 4 \ 32 & 12 \end{bmatrix} \\[1em] = \begin{bmatrix} 9 - 4 & 4 - 4 \ 32 - 32 & 17 - 12 \end{bmatrix} \\[1em] = \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix}.

Hence, X =[5005].\begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix}.

Now,

⇒X×[ab]=[550]⇒[5005]×[ab]=[550]⇒[5a+00a+5b]=[550].\Rightarrow X \times \begin{bmatrix} a \ b \end{bmatrix} = \begin{bmatrix} 5 \ 50 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix} \times \begin{bmatrix} a \ b \end{bmatrix} = \begin{bmatrix} 5 \ 50 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix} 5a + 0 \ 0a + 5b \end{bmatrix} = \begin{bmatrix} 5 \ 50 \end{bmatrix}.

Solving for a and b:

∴ 5a = 5

a = 55\dfrac{5}{5}

a = 1.

∴ 5b = 50

b = 505\dfrac{50}{5}

b = 10.

Hence, a = 1 and b = 10.

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