verify Cayley Hamilton theorem for the following matrix

July 10, 2021Posted inLinear AlgebraTags:
Question : Verify Cayley Hamilton theorem for the following matrix \[A=\left(\begin{matrix}1&-1&1\\2&-1&0\\1&0&0\end{matrix}\right) \] Solution : Let \[A=\left(\begin{matrix}1&-1&1\\2&-1&0\\1&0&0\end{matrix}\right)\] Then Characteristics equation of A is \[\left|A-xI\right|=0\] \[\left|\begin{matrix}1-x&-1&1\\2&-1-x&0\\1&0&-x\end{matrix}\right|=0\] \[\left(1-x\right)\left(-x\left(-1-x\right)-0\right)-\left(-1\right)\left(-2x-0\right)+1\left(-\left(-1-x\right)\right)=0\] \[x-x^3-2x+1+x=0\] \[x^3-1=0 \] In order…

How to find matrix power

July 10, 2021Posted inLinear AlgebraTags:
Introduction : Here we shall explain How to find matrix power through an example. Question : If \[A=\left(\begin{matrix}3&-4\\1&-1\end{matrix}\right)\] then find \(A^n\) Solution : Given matrix \[A=\left(\begin{matrix}3&-4\\1&-1\end{matrix}\right)\] second power of A…

solved prove that transpose(AB)=transpose(B)transpose(A) for the following matrices

July 9, 2021Posted inLinear AlgebraTags:
Question : Prove that transpose(AB)=transpose(B)transpose(A) for the following matrices Solution : Given two matrices \[A=\left(\begin{matrix}1&0&1\\0&1&1\end{matrix}\right)\ ,\ B=\left(\begin{matrix}1&4&3\\6&2&0\\-2&0&1\end{matrix}\right)\] Their transpose are \[A^T=\left(\begin{matrix}1&0\\0&1\\1&1\end{matrix}\right),\ B^T=\left(\begin{matrix}1&6&-2\\4&2&0\\3&0&1\end{matrix}\right)\] Product of A and B \[AB=\left(\begin{matrix}-1&4&4\\4&2&1\end{matrix}\right)\] Hence the…

Solved find the rank of the following matrix

July 9, 2021Posted inLinear AlgebraTags:
Question : Find row reduce form of the following matrix \[M=\left(\begin{matrix}3&-1&2&4\\9&-7&1&-2\end{matrix}\right)\] Solution : Given matrix \[M=\left(\begin{matrix}3&-1&2&4\\9&-7&1&-2\end{matrix}\right)\] To find rank of M Step 1: Applying elementary row transformation \(R_2\rightarrow R_2-3R_1\) \[M~\left(\begin{matrix}3&-1&2&4\\0&-4&-5&-14\\\end{matrix}\right)\]…