Matrices Linear Algebra Problem?
(a) Find a non-zero 2 × 2 matrix A such that A² has all zero entries?
(b) Prove or disprove: AX = AY implies X = Y , where A,X and Y are all 2 × 2 matrices.
(a) Find a non-zero 2 × 2 matrix A such that A² has all zero entries?
(b) Prove or disprove: AX = AY implies X = Y , where A,X and Y are all 2 × 2 matrices.
Comments
(a) Let A =
[0 1]
[0 0].
Then, A^2 is the zero matrix.
(Note:
[0 1]^2
[1 0] is the identity matrix.)
(b) This is false; let A and X be the matrix from part (a), and let Y = zero matrix.
Then, AX = AY = zero matrix, but X and Y are different.
I hope this helps!
(a)
A=
-1 1
-1 1
or
A=
-1 -1
1 1
(b)
If A is the zero 2 x 2 matrix then X and Y can be 2 different matrices. Although AX=AY this doesn't imply X=Y.
A is basically the inverse matrix of the 1st matrix. to discover inverse matrix, use unique matrix and improve it with id matrix: | a million 2 a million 0 | | 3 5 0 a million | Then cut back making use of Gauss-Jordan removal to get id matrix on left, making use of complication-loose row operations. you will get augmented matrix in this style: | a million 0 a b | | 0 a million c d | the place A= | a b | | c d | -------------------- Augmented matrix: | a million 2 a million 0 | | 3 5 0 a million | R2 = 3R1 - R2 = (3 6 3 0) - (3 5 0 a million) = (0 a million 3 -a million) | a million 2 a million 0 | | 0 a million 3 -a million| R1 = R1 - 2R2 = (a million 2 a million 0) - (0 2 6 -2) = (a million 0 -5 2) | a million 0 -5 2 | | 0 a million 3 -a million | A = | -5 2 | . . . .| 3 -a million | -------------------- verify: | a million 2 | . . | -5 2 | . .=. . | -5+6 .... 2-2 | . . . . | a million 0 | | 3 5 | . . | 3 -a million | . . . . .|-15+15.. 6-5 | . . . . | 0 a million | ok
(a) Choose A = a11 = 0, a12 = 1, a21 = 1, a22 = 0
(b) Choose X = a11 = 0, a12 = 2, a21 = 2, a22 = 0. Choose Y = 2X.
You'll see that AX = AY = 0, but X does not equal Y