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Related Lessons: Determinant of 3x3 Matrix

 

Determinant of 2x2 Matrix

 

Suppose we are given a square matrix A with four elements: a, b, c and d..

matrix A = [ [a b] [c d] ]

The determinant of matrix A is calculated as

det A = ad - bc

 

If you can't see the pattern yet, this is how it looks when the elements of the matrix are color coded.

  • We take the product of the elements from top left to bottom rightthen subtract by the product of the elements from top right to bottom left.

 

Determinant of 2 x 2 Matrix (animated)
animated image showing how to remember the formula of 2x2 determinant

 


 

Example 1: Find the determinant of matrix matrix A = [ [1  2] [3 4] ].

 

This is an example where all elements of the 2x2 matrix are positive.

det [ [1  2] [3 4] ] = 1*4-2*3=-2

 


 

Example 2: Find the determinant of matrix matrix B = [ [-5 -4] [ -2 -3] ].

 

Here is an example when all elements are negative. Make sure to apply the basic rules when multiplying integers. Remember, the product of numbers with the same signs will always be positive. In contrary, if the signs are different the product will be negative.

det  [ [-5 -4] [ -2 -3] ] = (-5*-3)-(-4*-2)=7

 


 

Example 3: Evaluate the determinant of matrix matrix C = [ [-1 -2] [6 3] ].

 

Make sure to remember the rule in subtracting numbers. That is, when you subtract, change the operation from subtraction to addition but you must switch the sign of the number directly found to its right. Other than that, proceed as usual.

det  [ [-1 -2] [6 3] ] = (-1*3)-(-2*6)=9

 


 

Example 4: Evaluate the determinant of matrix matrix D = [ [5 -3] [x y] ].

 

You may also encounter a problem where some of the elements in the matrix are variables. Treat this just like a normal determinant problem. Plug those variables in the designated spots in the formula then simplify as usual.

det [ [5 -3] [x y] ] = (5*y)-(-3*x)= 5y+3x

 


 

Example 5: Find the value of x in matrix F = [ [-4 2] [-8 x] ] if its determinant has a value of -12.

 

This is not a "trick" question. We can actually find the value of "x" such that when we apply the formula we get −12.

Get the determinant of the given matrix and set it equal to −16. By doing so, we generate a simple linear equation that is solvable for "x".

det [ [-4 2] [-8 x ] ] = -12, implies that x = 7

 

Checking our answer: Replace "x" by 7, then solve for the determinant. We expect to get −12.

determinant of [ [-4 2] [-8 7] ] = -12

This verifies that our solution is correct!

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

 

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