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Mathematics

If I is the unit matrix of order 2 × 2; find the matrix M such that :

(i) M - 2I = 3[−1041]3\begin{bmatrix}[r] -1 & 0 \ 4 & 1 \end{bmatrix}

(ii) 5M + 3I = 4[2−50−3]4\begin{bmatrix}[r] 2 & -5 \ 0 & -3 \end{bmatrix}

Matrices

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Answer

I = [1001]\begin{bmatrix}[r] 1 & 0 \ 0 & 1 \end{bmatrix}

(i) Given,

⇒M−2I=3[−1041]⇒M−2[1001]=3[−1041]⇒M−[2002]=[−30123]⇒M=[−30123]+[2002]⇒M=[−3+20+012+03+2]⇒M=[−10125].\Rightarrow M - 2I = 3\begin{bmatrix}[r] -1 & 0 \ 4 & 1 \end{bmatrix} \\[1em] \Rightarrow M - 2\begin{bmatrix}[r] 1 & 0 \ 0 & 1 \end{bmatrix} = 3\begin{bmatrix}[r] -1 & 0 \ 4 & 1 \end{bmatrix} \\[1em] \Rightarrow M - \begin{bmatrix}[r] 2 & 0 \ 0 & 2 \end{bmatrix} = \begin{bmatrix}[r] -3 & 0 \ 12 & 3 \end{bmatrix} \\[1em] \Rightarrow M = \begin{bmatrix}[r] -3 & 0 \ 12 & 3 \end{bmatrix} + \begin{bmatrix}[r] 2 & 0 \ 0 & 2 \end{bmatrix} \\[1em] \Rightarrow M = \begin{bmatrix}[r] -3 + 2 & 0 + 0 \ 12 + 0 & 3 + 2 \end{bmatrix} \\[1em] \Rightarrow M = \begin{bmatrix}[r] -1 & 0 \ 12 & 5 \end{bmatrix}.

Hence, M = [−10125].\begin{bmatrix}[r] -1 & 0 \ 12 & 5 \end{bmatrix}.

(ii) Given,

⇒5M+3I=4[2−50−3]⇒5M+3[1001]=[8−200−12]⇒5M+[3003]=[8−200−12]⇒5M=[8−200−12]−[3003]⇒5M=[8−3−20−00−0−12−3]⇒5M=[5−200−15]⇒M=15[5−200−15]⇒M=[1−40−3].\Rightarrow 5M + 3I = 4\begin{bmatrix}[r] 2 & -5 \ 0 & -3 \end{bmatrix} \\[1em] \Rightarrow 5M + 3\begin{bmatrix}[r] 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix}[r] 8 & -20 \ 0 & -12 \end{bmatrix} \\[1em] \Rightarrow 5M + \begin{bmatrix}[r] 3 & 0 \ 0 & 3 \end{bmatrix} = \begin{bmatrix}[r] 8 & -20 \ 0 & -12 \end{bmatrix} \\[1em] \Rightarrow 5M = \begin{bmatrix}[r] 8 & -20 \ 0 & -12 \end{bmatrix} - \begin{bmatrix}[r] 3 & 0 \ 0 & 3 \end{bmatrix} \\[1em] \\[1em] \Rightarrow 5M = \begin{bmatrix}[r] 8 - 3 & -20 - 0 \ 0 - 0 & -12 - 3 \end{bmatrix} \\[1em] \Rightarrow 5M = \begin{bmatrix}[r] 5 & -20 \ 0 & -15 \end{bmatrix} \\[1em] \Rightarrow M = \dfrac{1}{5}\begin{bmatrix}[r] 5 & -20 \ 0 & -15 \end{bmatrix} \\[1em] \Rightarrow M = \begin{bmatrix}[r] 1 & -4 \ 0 & -3 \end{bmatrix}.

Hence, M = [1−40−3].\begin{bmatrix}[r] 1 & -4 \ 0 & -3 \end{bmatrix}.

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