Find the value of x given that A2 = B where,
A=[21201] and B=[4x01].A = \begin{bmatrix}[r] 2 & 12 \ 0 & 1 \end{bmatrix} \text{ and } B = \begin{bmatrix}[r] 4 & x \ 0 & 1 \end{bmatrix}.A=[20121] and B=[40x1].
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Given A2=B,⇒[21201][21201]=[4x01]⇒[2×2+12×02×12+12×10×2+1×00×12+1×1]=[4x01]⇒[4+024+120+00+1]=[4x01]⇒[43601]=[4x01]\text{Given } A^2 = B, \\[1em] \Rightarrow \begin{bmatrix}[r] 2 & 12 \ 0 & 1 \end{bmatrix} \begin{bmatrix}[r] 2 & 12 \ 0 & 1 \end{bmatrix} = \begin{bmatrix}[r] 4 & x \ 0 & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 2 \times 2 + 12 \times 0 & 2 \times 12 + 12 \times 1 \ 0 \times 2 + 1 \times 0 & 0 \times 12 + 1 \times 1 \end{bmatrix} = \begin{bmatrix}[r] 4 & x \ 0 & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 4 + 0 & 24 + 12 \ 0 + 0 & 0 + 1 \end{bmatrix} = \begin{bmatrix}[r] 4 & x \ 0 & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 4 & 36 \ 0 & 1 \end{bmatrix} = \begin{bmatrix}[r] 4 & x \ 0 & 1 \end{bmatrix} \\[1em]Given A2=B,⇒[20121][20121]=[40x1]⇒[2×2+12×00×2+1×02×12+12×10×12+1×1]=[40x1]⇒[4+00+024+120+1]=[40x1]⇒[40361]=[40x1]
By definition of equality of matrices,
⇒ x = 36.
Hence, the value of x = 36.
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