2x2 Determinants
This page follows the JoeCMath video The Diagonal Method for 2x2 Determinants. The video starts with one numeric determinant, then rewrites the same idea as the general $ad-bc$ formula for a $2\times 2$ matrix.
Main Idea
For a $2\times 2$ matrix
\[A= \begin{bmatrix} a & b \\ c & d \end{bmatrix},\]the determinant is
\[\det(A)= \begin{vmatrix} a & b \\ c & d \end{vmatrix} =ad-bc.\]The video’s diagonal method is a way to keep that subtraction in the right order. The diagonal through $a$ and $d$ is the product that comes first. The diagonal through $b$ and $c$ is the product that gets taken away.
Watch this section: the general $2\times 2$ determinant formula at 0:30.
Opening Example
The first determinant in the video is
\[\begin{vmatrix} 2 & 5 \\ 3 & 9 \end{vmatrix}.\]The down-right diagonal gives
\[2\cdot 9=18.\]The other diagonal gives
\[5\cdot 3=15.\]Now subtract:
\[\begin{vmatrix} 2 & 5 \\ 3 & 9 \end{vmatrix} =2\cdot 9-5\cdot 3 =18-15 =3.\]Watch this section: the thumbnail example at 0:00.
Why The Order Matters
For
\[\begin{vmatrix} a & b \\ c & d \end{vmatrix},\]the answer is $ad-bc$, not $bc-ad$. The two products are the same products either way, but the subtraction order changes the sign.
That is why the video marks one diagonal first and then marks the other diagonal second. The first product is the positive product, and the second product is the product being subtracted.
Watch this section: connect the colored diagonals to $ad-bc$ at 0:30.
Practice With The Formula
The last part of the video uses the general formula on more $2\times 2$ determinants. The process stays the same:
\[\begin{aligned} \det(A) &= \begin{vmatrix} a & b \\ c & d \end{vmatrix} \\ &=ad-bc. \end{aligned}\]So the main thing to track is which entries play the roles of $a$, $b$, $c$, and $d$. Once the entries are in those four positions, the determinant is the first diagonal product minus the second diagonal product.
Watch this section: examples using the general formula at 0:55.
Where This Fits
The $2\times 2$ determinant is also the small calculation that shows up inside larger determinant work. When a $3\times 3$ minor leaves a $2\times 2$ determinant, this same $ad-bc$ rule finishes the minor.
For the next steps, use the Matrix Minors companion, the Matrix Cofactors companion, or the Cofactor Expansion companion.
Watch this section: the full 2-minute determinant walkthrough.
Timestamp Guide
For more JoeCMath linear algebra videos, use the linear algebra playlist.