What is determinant It gives a single value

What is determinant? It gives a single value associated with square matrix.

● Cramer’s Rule can be used to solve the system of equations in terms of Determinant Formula: = Coefficient determinant = Replace the first Column with answer matrix. = Replace the second Column with answer matrix. = Replace third Column with answer matrix. Instead of solving the entire system of equations, Cramer's can be used to solve for just one single variable

Determinant of a 2 x 2 matrix Finding Determinants of Matrices A= 3 2 Notice the different symbol: -5 4 the straight lines tell you to find the determinant!! = = = (3 * 4) 12 (-5 * 2) - (-10) 22 Slide 23 (0 f 56)

Determinant of a 3 x 3 matrix a 11 a 12 a 13 If A is a 3 x 3 matrix A= a 21 a 22 a 23 a 31 a 32 a 33 Then we define a 11 a 12 a 13 det(A)= a 21 a 22 a 23 = a 11 a 32 a 33 Example 1 det(A)= 1 5 a 22 a 23 a 32 a 33 -a 12 a 21 a 23 a 31 a 33 +a 13 a 21 a 22 a 31 a 32 -3 0 2 3 -1 2 =1 0 2 -1 2 -5 =1[0*2 - (-1*2)] – 5 [1*2 -2*3] - 3[-1*1 -0*3]=25 1 2 3 2 -3 1 0 3 -1

Application of system of equations using Cramer’s rule

The cost of 2 kg of wheat and 1 kg of sugar is RM 7. The cost of 1 kg wheat and 1 kg of rice is RM 7. The cost of 3 kg of wheat, 2 kg of sugar and 1 kg of rice is RM 17. Find the cost of rice per kg using Cramer’s Rule. 1. 2. Set a coefficient matrix Calculate a determinant of this matrix. 2(0 - 2) - 1 (1 – 3)] + 0[(2 – 0) = -4 + 2 = -2 3. Replace “Z” column of the main matrix by constant column and calculate its determinant. 2(0 - 14) - 1 (17 – 21)] + 7[(2 – 0) = -28 + 4 + 14 = -10