Matrices

Let's go through some questions and solutions related to matrices.

Question 1:

Matrices: Consider matrices 

$A=\left[\begin{array}{cc}2& -1\\ 3& 4\end{array}\right]$

and 

$B=\left[\begin{array}{cc}1& 0\\ -2& 5\end{array}\right]$

 

Question: Calculate the sum 

$C=A+B$

C=A+B.

Solution: Matrix addition is performed element-wise, so 

$C_{\mathrm{ij}\mathrm{<br>}}=A_{ij}+B_{\mathrm{ij}\mathrm{<br>}}$

  

$C=A+B=\left[\begin{array}{cc}2+1& -1+0\\ 3+(-2)& 4+5\end{array}\right]$

 

$C=\left[\begin{array}{cc}3& -1\\ 1& 9\end{array}\right]$

So, 

$C=\left[\begin{array}{cc}3& -1\\ 1& 9\end{array}\right]$

Question 2:

Matrices: Let 

$D=\left[\begin{array}{cc}4& -2\\ 1& 3\end{array}\right]$

Question: Calculate the transpose of matrix D

$\mathrm{<br>}$

Solution: The transpose of a matrix is obtained by swapping its rows with columns. 

$D^T=\left[\begin{array}{cc}4& 1\\ -2& 3\end{array}\right]$

 

So, the transpose of D is

$\mathrm{<br>}$ $D^T=\left[\begin{array}{cc}4& 1\\ -2& 3\end{array}\right]$

Question 3:

Matrices: Consider matrices 

$E=\left[\begin{array}{cc}2& 3\\ -1& 0\end{array}\right]$

E=[2−1​30​] and 

$F=\left[\begin{array}{cc}1& 2\\ 4& -3\end{array}\right]$

 

Question: Find the product 

$G=E\cdot F$

 

Solution: Matrix multiplication involves taking the dot product of each row of the first matrix with each column of the second matrix. 

$G_{\mathrm{ij}\mathrm{<br>}}={\sum }_{\mathrm{k=1}=1}^n{E}_{\mathrm{ik}\mathrm{<br>}}\cdot F_{kj}$

Gij​=∑k=1n​Eik​⋅Fkj​

Let's calculate each element of the resulting matrix G:

$\mathrm{<br>}$

$G_{11}=(2\cdot 1)+(3\cdot 4)=2+12=14$

 

$G_{12}=(2\cdot 2)+(3\cdot -3)=4-9=-5$

 

$G_{21}=(-1\cdot 1)+(0\cdot 4)=-1$

 

$G_{22}=(-1\cdot 2)+(0\cdot -3)=-2$

 

So, 

$E\cdot F=\left[\begin{array}{cc}14& -5\\ -1& -2\end{array}\right]$

 

These questions cover matrix addition, transpose, and multiplication. If you have more questions or if there's a specific aspect you'd like to explore, feel free to ask!