Given the following vector X, find a non-zero square matrix A such that AX=0:
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X= 2
-8
6
A= _ _ _
_ _ _
_ _ _
Please help, I do not understand this..
Oh haha that makes so much sense now! Thank you so much.
Well, it seems like you're in a bit of a matrix pickle! Don't worry, I'm here to help. So, you're looking for a non-zero square matrix A such that AX equals zero, right?
To find such a matrix, let's consider the equation AX = 0. The only way for the product of a matrix and a vector to be zero is if either the matrix or the vector (or both) consists of all zeroes. In this case, since we're looking for a non-zero matrix, let's set the vector X to all zeroes:
X = 0
0
0
Now, we need to find a non-zero matrix A such that AX = 0. Any non-zero matrix multiplied by a zero vector will give us a zero vector. So, for example, we can choose the identity matrix as A:
A = 1 0 0
0 1 0
0 0 1
And when we multiply this matrix A by the zero vector X, we get:
AX = 1*0 + 0*0 + 0*0
0*0 + 1*0 + 0*0
0*0 + 0*0 + 1*0
Which simplifies to:
AX = 0
0
0
Voila! We've found a non-zero square matrix A such that AX equals zero.
To find a non-zero square matrix A such that AX = 0, we need to solve the equation AX = 0 for the given vector X. In other words, we need to find a matrix A that, when multiplied with X, gives us the zero vector.
Let's start by multiplying the vector X with the matrix A:
AX = 0
A * X = 0
Let the matrix A be a 3x3 matrix with elements a, b, c, d, e, f, g, h, and i:
A = | a, b, c |
| d, e, f |
| g, h, i |
Now, we can multiply A with X:
A * X = | a*2 + b*(-8) + c*6 |
| d*2 + e*(-8) + f*6 |
| g*2 + h*(-8) + i*6 |
Simplifying the above expression, we get:
A * X = | 2a - 8b + 6c |
| 2d - 8e + 6f |
| 2g - 8h + 6i |
Since AX = 0, we need to set each element of the resulting matrix to 0:
2a - 8b + 6c = 0
2d - 8e + 6f = 0
2g - 8h + 6i = 0
Let's solve these equations to find the values of a, b, c, d, e, f, g, h, and i:
From the first equation, we can solve for a:
2a = 8b - 6c
a = 4b - 3c
From the second equation, we can solve for d:
2d = 8e - 6f
d = 4e - 3f
From the third equation, we can solve for g:
2g = 8h - 6i
g = 4h - 3i
Now, we can substitute the values of a, d, and g in terms of b, c, e, f, h, and i in the matrix A:
A = | 4b - 3c, b, c |
| 4e - 3f, e, f |
| 4h - 3i, h, i |
This is a non-zero square matrix A that satisfies AX = 0 for the given vector X.
or
1 1 1
2 2 2
3 3 3
:)
2 a11 - 8 a12 + 6 a13 = 0
let a11 = 1, a12 = 1 , a13 = 1
2a21 - 8 a22 + + 6 a23 = 0
again
a21 = a22 = a23
so try
1 1 1
1 1 1
1 1 1
:)