## Algebra - Matrices

### 3. Rank of a matrix

• Definition 3.1: A matrix which is obtained by leaving some rows or columns or both of a matrix is called sub-matrix of a matrix.

• Example 3.1: Let $A=\begin{bmatrix} 1&2&3&4\\5&6&7&8\\9&4&6&1 \end{bmatrix}$ then $\begin{bmatrix} 1&2&3&4\\5&6&7&8 \end{bmatrix}$, $\begin{bmatrix} 5&6&7&8\\9&4&6&1 \end{bmatrix}$, $\begin{bmatrix} 2&3&4\\6&7&8\\4&6&1 \end{bmatrix}$, $\begin{bmatrix} 6&7\\4&6 \end{bmatrix}$, $\begin{bmatrix} 7&8\\6&1 \end{bmatrix}$, $\begin{bmatrix} 1\\5\\9 \end{bmatrix}$, $\begin{bmatrix} 6&7 \end{bmatrix}$ etc. are sub-matrix of $A$.

• Definition 3.2: A matrix $A$ is said to have rank $r$ if there exists atleast one minor of order $r$ of the matrix $A$ which does not vanish and all the minors of order $r+1$ must vanish.
Notation: Rank of the matrix $A$ is denoted by $\rho(A)$

• Example 3.1: Find the rank of the matrix $A=\begin{bmatrix}1&2&3&1\\2&4&6&2\\1&2&3&2\end{bmatrix}$
Solution: Note that the possible rank of $A$ is 3 or 2 or 1. One can easily observe that the minor of order $3 \times 3$ is equal to zero. Thus the rank of $A$ is not equal to 3. There exist a minor of order $2 \times 2$ which is not equal to zero i.e, $\exists$ $\begin{vmatrix} 6&2\\3&2\end{vmatrix}$ which is not equal to zero. Hence rank of $A$ is 2. $\spadesuit$

• Example 3.2: Find the rank of $A=\begin{bmatrix} 1&2&5\\6&8&9\\1&3&6 \end{bmatrix}$
Solution: There exist a minor of order $3 \times 3$ which is not equal to zero i.e., $\begin{vmatrix} 1&2&5\\6&8&9\\1&3&6 \end{vmatrix} \neq 0$. Hence the rank of $A$ is 3. $\spadesuit$

• Another simply way to find the rank of a matrix is to bring the matrix to row reduced echelon form.

• Definition 3.3: A matrix is said to be in row reduced echelon form if
(i) The leading element (the first non zero element) of each row is one.
(ii) All the elements below the leading element is zero.
(iii) The number of zeros appering before the leading element in each row is greater than that appears in its previous row.
(iv) The zero rows must appear below the non zero rows.

• Any non zero matrix $A$ can be written in row reduced echelon form by applying a finite sequence of elementary row operations.

• The number of non zero rows in row reduced echelon form of a matrix is the rank of the matrix.

• Example 3.3: Find the row reduced echelon form of the matrix $\begin{bmatrix} 1&1&1&1\\1&2&3&4\\1&3&5&7\\1&4&7&10 \end{bmatrix}$ and hence find the rank.
Solution: Let $A=\begin{bmatrix} 1&1&1&1\\1&2&3&4\\1&3&5&7\\1&4&7&10 \end{bmatrix}$
$R_2\rightarrow R_2-R_1$, $R_3\rightarrow R_3-R_1$, $R_4\rightarrow R_4-R_1$
$\approxeq \begin{bmatrix} 1&1&1&1\\0&1&2&3\\0&2&4&6\\0&3&6&9 \end{bmatrix}$
$R_3\rightarrow R_3/2$, $R_4\rightarrow R_4/3$
$\approxeq \begin{bmatrix} 1&1&1&1\\0&1&2&3\\0&1&2&3\\0&1&2&3 \end{bmatrix}$
$R_3\rightarrow R_3-R_2$, $R_4\rightarrow R_4-R_2$
$\approxeq \begin{bmatrix} 1&1&1&1\\0&1&2&3\\0&0&0&0\\0&0&0&0 \end{bmatrix}$ is the row reduced echelon form of $A$. $\rho(A)=2$. $\spadesuit$

• Theorem 3.1: The rank of a matrix and its transpose are same.

• Theorem 3.2: The rank of the matrix is unaltered by the elementary operations i.e., equivalent matrices have same rank.

• Problem 3.1: Find the rank of $\begin{bmatrix} 1&2&3\\2&3&4\\0&0&2 \end{bmatrix}$
Solution: Consider $A=\begin{bmatrix} 1&2&3\\2&3&4\\0&0&2 \end{bmatrix}$
$R_2\rightarrow R_2-2R_1$
$\approxeq \begin{bmatrix} 1&2&3\\0&-1&-2\\0&0&2 \end{bmatrix}$
$R_2\rightarrow -R_2$; $R_3\rightarrow R_3/2$
$\approxeq \begin{bmatrix} 1&2&3\\0&1&2\\0&0&1 \end{bmatrix}$ $\rho(A)=$3. $\spadesuit$

• Problem 3.2: Find the rank of $\begin{bmatrix} 1&2&3\\4&5&6\\2&1&2 \end{bmatrix}$
Solution: Consider, $\begin{bmatrix} 1&2&3\\4&5&6\\2&1&2 \end{bmatrix}$
$R_2\rightarrow R_2-4R_1$; $R_3\rightarrow R_3-2R_1$
$\approxeq \begin{bmatrix} 1&2&3\\0&-3&-6\\0&-3&-4 \end{bmatrix}$
$R_2\rightarrow R_2/-3$
$\approxeq \begin{bmatrix} 1&2&3\\0&1&2\\0&-3&-4 \end{bmatrix}$
$R_3\rightarrow R_3+2R_2$
$\approxeq \begin{bmatrix} 1&2&3\\0&1&2\\0&0&2 \end{bmatrix}$
$R_3\rightarrow R_3/2$
$\approxeq \begin{bmatrix} 1&2&3\\0&1&2\\0&0&1 \end{bmatrix}$ $\rho(A)=3$. $\spadesuit$
• Problem 3.3: Find the rank of $\begin{bmatrix} 1&2&3&2\\2&3&5&1\\1&3&4&5 \end{bmatrix}$
Solution: Consider, $\begin{bmatrix} 1&2&3&2\\2&3&5&1\\1&3&4&5 \end{bmatrix}$
$R_2\rightarrow R_2-2R_1$; $R_3\rightarrow R_3-R_1$
$\approxeq \begin{bmatrix} 1&2&3&2\\0&-1&-1&3\\0&1&1&3 \end{bmatrix}$
$R_2\rightarrow -R_2$
$\approxeq \begin{bmatrix} 1&2&3&2\\0&1&1&3\\0&1&1&3 \end{bmatrix}$
$R_3\rightarrow R_3-R_2$
$\approxeq \begin{bmatrix} 1&2&3&2\\0&1&1&3\\0&0&0&0 \end{bmatrix}$ $\rho(A)=2$. $\spadesuit$

• Problem 3.4: Find the rank of $\begin{bmatrix} 1&-3&1&2\\0&1&2&3\\3&4&1&-2 \end{bmatrix}$
Solution: Consider, $\begin{bmatrix} 1&-3&1&2\\0&1&2&3\\3&4&1&-2 \end{bmatrix}$
$R_3\rightarrow R_3-3R_1$
$\approxeq \begin{bmatrix} 1&-3&1&2\\0&1&2&3\\0&13&-2&-8 \end{bmatrix}$
$R_3\rightarrow R_3-13R-1$
$\approxeq \begin{bmatrix} 1&-3&1&2\\0&1&2&3\\0&0&-28&-47 \end{bmatrix}$
$R_3\rightarrow R_3/-28$
$\approxeq \begin{bmatrix} 1&-3&1&2\\0&1&2&3\\0&0&1&47/28 \end{bmatrix}$ $\rho(A)=2$. $\spadesuit$

• Problem 3.5: Find the rank of $\begin{bmatrix} 1&2&-3&-4\\1&3&1&-2\\2&5&2&-5 \end{bmatrix}$
Solution: Consider, $A=\begin{bmatrix} 1&2&-3&-4\\1&3&1&-2\\2&5&2&-5 \end{bmatrix}$
$R_2\rightarrow R_2-R_1$; $R_3\rightarrow R_3-2R_1$
$\approxeq$ $\begin{bmatrix} 1&2&-3&-4\\0&1&4&2\\0&1&8&3 \end{bmatrix}$
$R_3\rightarrow R_3-R_2$
$\approxeq$ $\begin{bmatrix} 1&2&-3&-4\\0&1&4&2\\0&0&4&1 \end{bmatrix}$ $\rho(A)=$3 $\spadesuit$

• Problem 3.6: Find the rank of $\begin{bmatrix} 1&-1&3&2\\1&3&-3&4\\5&3&3&9 \end{bmatrix}$
Solution: Consider, $A=\begin{bmatrix} 1&-1&3&2\\1&3&-3&4\\5&3&3&9 \end{bmatrix}$
$R_2\rightarrow R_2-R_1$; $R_3\rightarrow R_3-5R_1$
$\approxeq$ $\begin{bmatrix} 1&-1&3&2\\0&4&-6&2\\0&8&-12&-1 \end{bmatrix}$
$R_3\rightarrow R_3-2R_1$
$\approxeq$ $\begin{bmatrix} 1&-1&3&2\\0&4&-6&2\\0&0&0&-5 \end{bmatrix}$ $\rho(A)=$3 $\spadesuit$

• Problem 3.7: Find the rank of $\begin{bmatrix} 3&-1&-1&2\\2&2&1&-1\\1&-3&2&1\\1&-3&6&-1\end{bmatrix}$
Solution: Consider, $A=\begin{bmatrix} 3&-1&-1&2\\2&2&1&-1\\1&-3&2&1\\1&-3&6&-1\end{bmatrix}$
$R_3\leftrightarrow R_1$
$\approxeq$ $\begin{bmatrix} 1&-3&2&1\\2&2&1&-1\\3&-1&-1&2\\1&-3&6&-1\end{bmatrix}$
$R_2\rightarrow R_2-2R_1$; $R_3\rightarrow R_3-3R_1$; $R_4\rightarrow R_4-R_1$
$\approxeq$ $\begin{bmatrix} 1&-3&2&1\\0&8&-3&-3\\0&8&-7&-1\\0&0&4&-2\end{bmatrix}$
$R_3\rightarrow R_3-R_2$
$\approxeq$ $\begin{bmatrix} 1&-3&2&1\\0&8&-3&-3\\0&0&-4&2\\0&0&4&-2\end{bmatrix}$
$R_4\rightarrow R_3+R_3$
$\approxeq$ $\begin{bmatrix} 1&-3&2&1\\0&8&-3&-3\\0&0&-4&2\\0&0&0&0\end{bmatrix}$ $\rho(A)=$3 $\spadesuit$

• Problem 3.8: Find the rank of $\begin{bmatrix} 1&0&2&-2\\2&-1&0&-1\\1&0&2&-1\\4&-1&3&-1 \end{bmatrix}$
Solution: Consider, $A=\begin{bmatrix} 1&0&2&-2\\2&-1&0&-1\\1&0&2&-1\\4&-1&3&-1 \end{bmatrix}$
$R_2\rightarrow R_2-2R_1$; $R_3\rightarrow R_3-R_1$; $R_4\rightarrow R_4-4R_1$
$\approxeq \begin{bmatrix} 1&0&2&-2\\0&-1&-4&1\\0&0&0&1\\0&-1&-6&-7 \end{bmatrix}$
$R_2\rightarrow \times -1$
$\approxeq \begin{bmatrix} 1&0&2&-2\\0&1&4&-1\\0&0&0&1\\0&-1&-6&-7 \end{bmatrix}$
$R_4\rightarrow R_4+R_2$
$\approxeq \begin{bmatrix} 1&0&2&-2\\0&1&4&-1\\0&0&0&1\\0&0&-2&-8 \end{bmatrix}$
$R_4\rightarrow R_4/-2$
$\approxeq \begin{bmatrix} 1&0&2&-2\\0&1&4&-1\\0&0&0&1\\0&0&1&4 \end{bmatrix}$
$R_3\leftrightarrow R_4$
$\approxeq \begin{bmatrix} 1&0&2&-2\\0&1&4&-1\\0&0&1&4\\0&0&0&1 \end{bmatrix}$ $\rho(A)=$4 $\spadesuit$

• Problem 3.9: Find the rank of $\begin{bmatrix} 2&4&3&4\\1&2&-1&4\\1&2&3&4\\-1&-2&6&-7 \end{bmatrix}$
Solution: Consider, $A=\begin{bmatrix} 2&4&3&4\\1&2&-1&4\\1&2&3&4\\-1&-2&6&-7 \end{bmatrix}$
$R_3\leftrightarrow R_1$
$\approxeq \begin{bmatrix} 1&2&3&4\\1&2&-1&4\\2&4&3&4\\-1&-2&6&-7 \end{bmatrix}$
$R_2\rightarrow R_2-R_1$; $R_3\rightarrow R_3-2R_1$; $R_4\rightarrow R_4+R_1$
$\approxeq \begin{bmatrix} 1&2&3&4\\0&0&-4&0\\0&0&3&-4\\0&0&9&-3 \end{bmatrix}$
$R_2\rightarrow R_2/-4$
$\approxeq \begin{bmatrix} 1&2&3&4\\0&0&1&0\\0&0&3&-4\\0&0&9&-3 \end{bmatrix}$
$R_3\rightarrow R_3-3R_2$; $R_4\rightarrow R_4-9R_2$
$\approxeq \begin{bmatrix} 1&2&3&4\\0&0&1&0\\0&0&0&-4\\0&0&0&-3 \end{bmatrix}$
$R_3\rightarrow R_3/-4$; $R_4\rightarrow R_4/-3$
$\approxeq \begin{bmatrix} 1&2&3&4\\0&0&1&0\\0&0&0&1\\0&0&0&1 \end{bmatrix}$
$R_4\rightarrow R_4-R_3$
$\approxeq \begin{bmatrix} 1&2&3&4\\0&0&1&0\\0&0&0&1\\0&0&0&0 \end{bmatrix}$ $\rho(A)=$3 $\spadesuit$

• Problem 3.10: Find the row reduced echelon form of $\begin{bmatrix} 1&1&1&1\\1&2&3&4\\1&3&5&7\\1&4&7&10 \end{bmatrix}$ and hence find the rank.
Solution: Let $A=\begin{bmatrix} 1&1&1&1\\1&2&3&4\\1&3&5&7\\1&4&7&10 \end{bmatrix}$
$R_2\rightarrow R_2-R_1$, $R_3\rightarrow R_3-R_1$, $R_4\rightarrow R_4-R_1$
$\approxeq \begin{bmatrix} 1&1&1&1\\0&1&2&3\\0&2&4&6\\0&3&6&9 \end{bmatrix}$
$R_3\rightarrow R_3/2$, $R_4\rightarrow R_4/3$
$\approxeq \begin{bmatrix} 1&1&1&1\\0&1&2&3\\0&1&2&3\\0&1&2&3 \end{bmatrix}$
$R_3\rightarrow R_3-R_2$, $R_4\rightarrow R_4-R_2$
$\approxeq \begin{bmatrix} 1&1&1&1\\0&1&2&3\\0&0&0&0\\0&0&0&0 \end{bmatrix}$ is the row reduced echelon form of $A$. $\rho(A)=2$. $\spadesuit$

• Problem 3.11: Find the rank of $\begin{bmatrix} 2&1&5&4\\3&-2&2&-4\\5&-8&-4&2 \end{bmatrix}$
Solution: Consider, $A=\begin{bmatrix} 2&1&5&4\\3&-2&2&-4\\5&-8&-4&2 \end{bmatrix}$
$R_2\rightarrow R_2-R_1$;
$\approxeq \begin{bmatrix} 2&1&5&4\\1&-3&-3&-8\\5&-8&-4&2 \end{bmatrix}$
$R_2\leftrightarrow R_1$
$\approxeq \begin{bmatrix} 1&-3&-3&-8\\2&1&5&4\\5&-8&-4&2 \end{bmatrix}$
$R_2\rightarrow R_2-2R_1$; $R_3\rightarrow R_3-5R_1$
$\approxeq \begin{bmatrix} 1&-3&-3&-8\\0&7&11&20\\0&7&11&42 \end{bmatrix}$
$R_3 \rightarrow R_3-R_2$
$\approxeq \begin{bmatrix} 1&-3&-3&-8\\0&7&11&20\\0&0&0&22 \end{bmatrix}$ $\rho(A)=$3 $\spadesuit$

• Problem 3.12: Find the rank of $\begin{bmatrix} 2&3&0&4\\3&2&4&0\\0&4&2&3\\4&0&3&2 \end{bmatrix}$
Solution: Consider $A=\begin{bmatrix} 2&3&0&4\\3&2&4&0\\0&4&2&3\\4&0&3&2 \end{bmatrix}$
$R_2 \rightarrow R_2-R_1$
$\approxeq \begin{bmatrix} 2&3&0&4\\1&-1&4&-4\\0&4&2&3\\4&0&3&2 \end{bmatrix}$
$R_2 \leftrightarrow R_1$
$\approxeq \begin{bmatrix} 1&-1&4&-4\\2&3&0&4\\0&4&2&3\\4&0&3&2 \end{bmatrix}$
$R_2 \rightarrow R_2-2R_1$; $R_3\rightarrow R_3-4R_1$
$\approxeq \begin{bmatrix} 1&-1&4&-4\\0&5&-8&12\\0&4&2&3\\0&4&-13&18 \end{bmatrix}$
$R_2 \rightarrow R_2-R_3$
$\approxeq \begin{bmatrix} 1&-1&4&-4\\0&1&-10&9\\0&4&2&3\\0&4&-13&18 \end{bmatrix}$
$R_3\rightarrow R_3-4R_2$; $R_4\rightarrow R_4-4R_2$
$\approxeq \begin{bmatrix} 1&-1&4&-4\\0&1&-10&9\\0&0&42&-33\\0&0&-27&-18 \end{bmatrix}$
$R_3\rightarrow R_3/42$
$\approxeq \begin{bmatrix} 1&-1&4&-4\\0&1&-10&9\\0&0&1&-11/14\\0&0&-27&-18 \end{bmatrix}$
$R_4\rightarrow R_4+27R_3$
$\approxeq \begin{bmatrix} 1&-1&4&-4\\0&1&-10&9\\0&0&1&-11/14\\0&0&0&-549/14 \end{bmatrix}$
$R_4\rightarrow R_4\times -14/549$
$\approxeq \begin{bmatrix} 1&-1&4&-4\\0&1&-10&9\\0&0&1&-11/14\\0&0&0&1 \end{bmatrix}$ $\rho(A)$=4 $\spadesuit$

• Problem 3.13: Find the rank of $\begin{bmatrix} 2&0&1&3\\1&3&2&0\\1&4&3&2\\3&2&1&4 \end{bmatrix}$
Solution: Consider $A=\begin{bmatrix} 2&0&1&3\\1&3&2&0\\1&4&3&2\\3&2&1&4 \end{bmatrix}$
$R_2\leftrightarrow R_1$
$\approxeq\begin{bmatrix} 1&3&2&0\\2&0&1&3\\1&4&3&2\\3&2&1&4 \end{bmatrix}$
$R_2\rightarrow R_2-2R_1$; $R_3\rightarrow R_3-R_1$; $R_4\rightarrow R_4-3R_1$
$\approxeq\begin{bmatrix} 1&3&2&0\\0&-6&-3&3\\0&1&1&2\\0&-7&-5&4 \end{bmatrix}$
$R_2\leftrightarrow R_3$
$\approxeq\begin{bmatrix} 1&3&2&0\\0&1&1&2\\0&-6&-3&3\\0&-7&-5&4 \end{bmatrix}$
$R_3\rightarrow R_3+6R_2$; $R_4\rightarrow R_4+7R_2$
$\approxeq\begin{bmatrix} 1&3&2&0\\0&1&1&2\\0&0&9&15\\0&0&2&18 \end{bmatrix}$
$R_3\rightarrow R_3/3$; $R_4\rightarrow R-4/2$
$\approxeq\begin{bmatrix} 1&3&2&0\\0&1&1&2\\0&0&3&5\\0&0&1&9 \end{bmatrix}$
$R_3 \leftrightarrow R_4$
$\approxeq\begin{bmatrix} 1&3&2&0\\0&1&1&2\\0&0&1&9\\0&0&3&5 \end{bmatrix}$
$R_4\rightarrow R_4-3R_3$
$\approxeq\begin{bmatrix} 1&3&2&0\\0&1&1&2\\0&0&1&9\\0&0&0&-22 \end{bmatrix}$
$R_4 \rightarrow R_4/-22$
$\approxeq\begin{bmatrix} 1&3&2&0\\0&1&1&2\\0&0&1&9\\0&0&0&1 \end{bmatrix}$ $\rho(A)=4$ $\spadesuit$

• Problem 3.14: Find the value of $a$ in order that the rank of the matrix $A$ is 3 where $A=\begin{bmatrix} 1&1&-1&0\\4&4&-3&1\\a&2&2&2\\9&9&a&3 \end{bmatrix}$
Solution: Consider $A=\begin{bmatrix} 1&1&-1&0\\4&4&-3&1\\a&2&2&2\\9&9&a&3 \end{bmatrix}$
$R_2 \rightarrow R_2-4R_1$; $R_3\rightarrow R_3-aR_1$; $R_4\rightarrow R_4-9R_1$
$\approxeq\begin{bmatrix} 1&1&-1&0\\0&0&1&1\\0&2-a&2+a&2\\0&0&a+9&3 \end{bmatrix}$
$R_3\leftrightarrow R_2$
$\approxeq\begin{bmatrix} 1&1&-1&0\\0&2-a&2+a&2\\0&0&1&1\\0&0&a+9&3 \end{bmatrix}$
$R_4\rightarrow R_4-3R_3$
$\approxeq\begin{bmatrix} 1&1&-1&0\\0&2-a&2+a&2\\0&0&1&1\\0&0&a+6&0 \end{bmatrix}$
Since the rank of $A$ is 3, $a+6=0$ $\Rightarrow$ $a=-6$ $\spadesuit$

• Problem 3.15: Find the value of $a$ in order that the rank of the matrix $A$ is 2 where $A=\begin{bmatrix} 6&a&-1\\2&3&1\\3&4&2 \end{bmatrix}$
Solution: Consider $A=\begin{bmatrix} 6&a&-1\\2&3&1\\3&4&2 \end{bmatrix}$
$R_3\rightarrow R_3-R_2$
$\approxeq\begin{bmatrix} 6&a&-1\\2&3&1\\1&1&1 \end{bmatrix}$
$R_3\leftrightarrow R_1$
$\approxeq\begin{bmatrix} 1&1&1\\2&3&1\\6&a&-1 \end{bmatrix}$
$R_2\rightarrow R_2-2R_1$; $R_3\rightarrow R_3-6R_1$
$\approxeq\begin{bmatrix} 1&1&1\\0&1&-1\\0&a-6&-7 \end{bmatrix}$
$R_3\rightarrow R_3-7R_2$
$\approxeq\begin{bmatrix} 1&1&1\\0&1&-1\\0&a-13&0 \end{bmatrix}$
Since the rank of $A$ is 2, $a-13=0$ $\Rightarrow$ $a=13$ $\spadesuit$

• Exercise 3.1: Find the rank of $\begin{bmatrix} 8&0&0&1\\1&0&8&1\\0&0&1&8\\0&8&1&8 \end{bmatrix}$

• Exercise 3.2: Find the rank of $\begin{bmatrix} 2&3&4&5\\1&2&3&4\\1&1&1&1\end{bmatrix}$

• Exercise 3.3: Find the rank of $\begin{bmatrix} 1&2&-1&4\\2&4&3&4\\1&2&3&4\\-1&-2&6&-7 \end{bmatrix}$

• Exercise 3.4: Find the rank of $\begin{bmatrix} 1&2&-2&3\\2&5&4&7\\-1&-3&2&-1\\2&4&-1&3 \end{bmatrix}$

• Exercise 3.5: Find the rank of $\begin{bmatrix} 1&2&-4&3\\2&-1&-3&5\\-1&8&-6&-1 \end{bmatrix}$

• Exercise 3.6: Find the value of $a$ in order that the matrix $A$ is of rank 2 where $A=\begin{bmatrix} 6&a&-1\\2&3&1\\3&4&2\end{bmatrix}$

• Exercise 3.7: Find the value of $a$ in order that the matrix $A$ is of rank 2 where $A=\begin{bmatrix} 2&4&-4&a\\-1&-2&-1&2\\1&2&-1&3 \end{bmatrix}$

• Exercise 3.8: If $A=\begin{bmatrix} 1&1&-1\\2&-3&4\\3&-2&3\end{bmatrix}$ and $B=\begin{bmatrix} -1&-2&-1\\6&12&6\\5&10&5 \end{bmatrix}$ find the rank of $AB$, $BA$ and $A+B$

• Exercise 3.9: Find the rank of $AB$ if $A=\begin{bmatrix} 1&2&3\\2&3&4\\0&2&2 \end{bmatrix}$ and $B=\begin{bmatrix} 2&3&0\\3&1&2\\-1&2&2 \end{bmatrix}$

• Exercise 3.10: If $A=\begin{bmatrix} 0&1&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0 \end{bmatrix}$ find the rank of $A$ and $A^2$