DETERMINANTS Dr. Shildneck Fall, 2015

What is a DETERMINANT? The determinant of a matrix is a NUMBER that is associated to that matrix that helps us to determine things about that matrix. Only SQUARE matrices have determinants. You will be required to find determinants of 2x2 and 3x3 determinants by hand.

NOTATIONS for DETERMINANTS A determinant might be asked for in a couple of different ways. “DET” form “Bar” form

“DET” form A = 1 2 3 4 Det 1 2 3 4 Det(A) or Given matrix The determinant is indicated by Det 1 2 3 4 Det(A) or

“BAR” form A = 1 2 3 4 Given matrix The determinant is also indicated by 1 2 3 4 |A| or

The DETERMINANT of a 2x2 Matrix To find the determinant of a 2x2 matrix use the “Down – Up” Diagonal Method. det 𝑎 𝑏 𝑐 𝑑 = ad - cb

The DETERMINANT of a 2x2 Matrix Find the value of the determinant det 1 2 3 4

The DETERMINANT of a 2x2 Matrix Find the value of the determinant 7 −3 8 2

The DETERMINANT of a 3x3 Matrix To find the determinant of a 3x3 matrix we will use the LATTICE METHOD… This method ultimately turns into an augmented version of the “Down – Up” Diagonal Method.

The LATTICE METHOD Copy the first and second column to the right of the matrix/determinant. Draw “Down” diagonals under each of the three 3-term “down” diagonals. Multiply the numbers in each diagonal and add them together… this is your “DOWN” total. Draw “Up” diagonals under each of the three 3-term “Up” diagonals. Multiply the numbers in each diagonal and add them together… this is your “UP” total. Now do “Down minus Up.”

The LATTICE METHOD 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 𝑎 𝑏 𝑑 𝑒 𝑔 ℎ = - ( ) aei +bfg +cdh = - ( ) aei +bfg +cdh ( ) gec +hfa +idb

The LATTICE METHOD 2 3 0 1 2 1 4 2 −1 2 3 1 2 4 2

DETERMINANT EQUATIONS Since Determinants are just numbers, they can be equal to some value, even if there are variables inside. To solve a variable equation, just evaluate the determinant using the processes we have discussed and simplify the algebraic expression. Then, set that expression equal to the value of the determinant and solve for the variable.

The DETERMINANT EQUATION Find the value of the unknown. 𝑥 5 1 −3 =16

ASSIGNMENT ASSIGNMENT # 6 - DETERMINANTS