baby matrix

[logistic_guy]

Evaluate the determinant of the given matrix by $\displaystyle \bold{(a)}$ cofactor expansion and $\displaystyle \bold{(b)}$ using elementary row operations to introduce zeros into the matrix.

$\displaystyle \bold{1.} \ \begin{bmatrix}-4 & 2 \\3 & 3 \end{bmatrix}$

$\displaystyle \bold{2.} \ \begin{bmatrix}7 & -1 \\-2 & -6 \end{bmatrix}$

 

[logistic_guy]

What is cofactor?

If we have a square matrix, say $\displaystyle A = [a_{ij}]$, then $\displaystyle C_{ij}$ is the cofactor of the entry $\displaystyle a_{ij}$.

And $\displaystyle C_{ij}$ is defined as:

$\displaystyle C_{ij} = (-1)^{i+j}M_{ij}$, where $\displaystyle M_{ij}$ is the minor and it is equal to the determinant of $\displaystyle (n - 1) \times (n - 1)$ matrix by deleting the $\displaystyle i$th row and $\displaystyle j$th column of $\displaystyle A$.

Confused? One example and you'll be the master of cofactor expansion

$\displaystyle \bold{1.} \rightarrow \ \bold{(a)}$

We can work along the $\displaystyle i$th row or along the $\displaystyle j$th column. Let us make a decision. We will work along $\displaystyle i$th row. We will choose to work along the first row, so

$\displaystyle \text{det} \ A = \left| \begin{matrix}-4 & 2 \\3 & 3\end{matrix} \right| = -4C_{11} + 2C_{12}$

$\displaystyle C_{11} = (-1)^{1+1}M_{11} = M_{11}$

$\displaystyle M_{11} \rightarrow$ means delete the first row and delete the first column and take the determinant of what left.

Then,

$\displaystyle M_{11} = |3| = 3$
$\displaystyle C_{11} = 3$

And

$\displaystyle C_{12} = (-1)^{1+2}M_{12} = -M_{12}$

$\displaystyle M_{12} \rightarrow$ means delete the first row and delete the second column and take the determinant of what left.

Then,

$\displaystyle M_{12} = |3| = 3$
$\displaystyle C_{12} = -3$

Finally, we have:

$\displaystyle \text{det} \ A = -4C_{11} + 2C_{12} = -4(3) + 2(-3) = -12 - 6 = -18$

Easy ha?I told you!

 

[logistic_guy]

$\displaystyle \bold{2} \rightarrow \bold{(a)}$

$\displaystyle \left| \begin{matrix}7 & -1 \\-2 & -6\end{matrix} \right| = 7C_{11} + (-2)C_{21} = 7(-1)^2M_{11} - 2(-1)^{3}M_{21} = 7|-6| + 2|-1| = -42 - 2 = -44$

 

[logistic_guy]

$\displaystyle \bold{1} \rightarrow \bold{(b)}$

Let us start with the original matrix.

$\displaystyle \begin{bmatrix}-4 & 2 \\3 & 3 \end{bmatrix}$

$\displaystyle R_2 \rightarrow \frac{3}{4}R_1 + R_2$

$\displaystyle \begin{bmatrix}-4 & 2 \\0 & 4.5 \end{bmatrix}$

We introduced a zero in the matrix. Now let us calculate the determinant.

$\displaystyle \left| \begin{matrix}-4 & 2 \\0 & 4.5\end{matrix} \right| = (-4)(4.5) - (0)(2) = -18$

 

[logistic_guy]

$\displaystyle \bold{2} \rightarrow \bold{(b)}$

Let us start with the original matrix.

$\displaystyle \begin{bmatrix}7 & -1 \\-2 & -6 \end{bmatrix}$

$\displaystyle R_1 \rightarrow \frac{7}{2}R_2 + R_1$

$\displaystyle \begin{bmatrix}0 & -22 \\-2 & -6 \end{bmatrix}$

We introduced a zero in the matrix. Now let us calculate the determinant.

$\displaystyle \left| \begin{matrix}0 & -22 \\-2 & -6\end{matrix} \right| = (0)(-6) - (-2)(-22) = -44$