Algebra Lineal

1.- Sean las matrices

1 0 2 1 -1 2 -1 3 2
A= 2 1 -1 B= 1 3 3 C= 1 2 2
0 0 2 -1 1 1 2 0 1


Se pide calcular
1. 5A + 2B -3C
Sol:
1 02 5 0 10
5 A = 5 * 2 1 -1 = 10 5 -5
0 0 2 0 0 10

1 -1 2 2 -2 4
2 B = 2 * 1 3 3 = 2 6 6
-1 1 1-2 2 2

-1 3 2 -3 9 6
3 A = 3* 1 2 2 = 3 6 6
2 0 1 6 0 3



5 + 2 - - 3 0 + - 2 - 910 + 4 - 6
A+B-C = 10 + 2 - 3 5 + 6 - 6 - 5 + 6 - 6
0 + - 2 – 6 0 + 2 - 0 10 + 2 - 3




10 - 11 8
A+B-C = 95 - 5
- 8 2 9



2.- BT + C A =


-1 3 2 1 0 2
C= 1 2 2 A= 2 1 -1
2 0 1 0 0 2-1 x 1 + 3 x 2 + 2 x 0 = 5 -1 x 0 + 3 x 1 + 2 x 0 = 3 -1 x 2 + 3 x -1 + 2 x 2 = -1
1 x 1 + 2 x 2 + 2 x 0 = 5 1 x 0 + 2 x 1 + 2 x 0 = 2 1 x 2 + 2 x -1 + 2 x 2 = 4
2 x 1 + 0 x 2 + 1 x 0 = 22 x 0 + 0 x 1 + 1 x 0 = 0 2 x 2 + 0 x -1 + 1 x 2 = 6


5 3 - 1 1 1 -1
CA= 5 2 4 BT = -1 3 1
2 0 6 2 3 1



1+5 1+3 -1+-1 6 4 -2
BT+CA= -1+5 3+21+4 = 4 5 5
2+2 3+0 1+6 4 3 7






3.- Rango de la Matriz A

Sol.-

1.- Sean las matrices
1 0 2 1 0 2
A= 2 1 -1 F2 – 2F1 → F2 B = 0 1 -3 => Ran(A) =3
0 0 2 0 0 2





4.- Inversa de A (usando Gauss o Cofactores)


1 0 2
A= 2 1 -1
0 0 2



1 0 2 1 0 2
2 1 -1 2 1 -2
0 02 2 0 0 0 2
1 0 2 0 0 1 0 2
2 1 -1 0 0 2 1 -1


(A) = (2+0+0)-(0) = 2



C11 = 1 -1
0 2 = 1*2 - -1*0 = 2-0 =2...