Diagonal Method for 3x3 Matrices
Master the Diagonal Method for 3x3 determinants: copy two columns, multiply along three downward diagonals, subtract the three upward diagonal products for a fast, visual shortcut.
Key Concepts
A second method for a $3 \times 3$ determinant involves repeating the first two columns to the right of the matrix. Add the products of the three downward sloping diagonals, and subtract the sum of the products of the three upward sloping diagonals.
Evaluate $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}$. Rewrite: $\begin{array}{ccccc} 1 & 2 & 3 & 1 & 2 \\ 4 & 5 & 6 & 4 & 5 \\ 7 & 8 & 9 & 7 & 8 \end{array}$. Result: $((45)+(84)+(96)) ((105)+(48)+(72)) = 225 225 = 0$. Evaluate $\begin{vmatrix} 3 & 1 & 2 \\ 0 & 5 & 3 \\ 1 & 4 & 2 \end{vmatrix}$. Rewrite: $\begin{array}{ccccc} 3 & 1 & 2 & 3 & 1 \\ 0 & 5 & 3 & 0 & 5 \\ 1 & 4 & 2 & 1 & 4 \end{array}$. Result: $(( 30)+( 3)+0) (10+( 36)+0) = 33 ( 26) = 7$.
This is a fantastic visual shortcut for 3x3 determinants, no minors needed! First, copy the first two columns and place them next to the matrix. Then, multiply along the three downward diagonals and add their products. Do the same for the three upward diagonals. Finally, subtract the total of the 'up' products from the 'down' products.
Common Questions
What is the Diagonal Method for a 3x3 matrix determinant?
The Diagonal Method repeats the first two columns to the right of the 3x3 matrix. You multiply along three downward-sloping diagonals and add those products, then multiply along three upward-sloping diagonals and add those. The determinant equals the downward sum minus the upward sum.
How do you avoid sign errors with the Diagonal Method?
The most common mistake is adding the two sums instead of subtracting. Always write the formula as (Downhill Sum) minus (Uphill Sum) before you start, and be careful with negative signs during each diagonal multiplication.
When should Grade 10 students use the Diagonal Method vs cofactor expansion?
The Diagonal Method is a fast visual shortcut exclusive to 3x3 matrices. Cofactor expansion works for any size matrix and reinforces the underlying structure, so use both methods to verify answers during practice.