A is a 2x3 matrix in RREF, and vector v is a solution to the homogeneous equation Av = 0, where v = (1, -2, -1). Which of the following matrices could A be equal to?
Solution
The question states that matrix A is a 2×3 matrix in row-reduced echelon form (RREF), and vector v=1−2−1 is a solution to the homogeneous equation Av=0.
We are tasked with identifying which of the three matrices given could be equal to A.
Step-by-step approach:
Since A is in RREF and v is a solution to Av=0, when we multiply the matrix A by the vector v, we must get the zero vector.
The matrix A is 2×3, and the vector v has 3 components. This confirms the multiplication Av=0 is valid.
Matrix A options:
(i) A=(10011−2)
(ii) A=(1001−12)
(iii) A=(100112)
Multiplying each matrix by v=1−2−1 to check if Av=0 holds.
Testing Option (i):
A=(10011−2),v=1−2−1Av=(1⋅1+0⋅(−2)+1⋅(−1)0⋅1+1⋅(−2)+(−2)⋅(−1))=(1−1−2+2)=(00)
This is correct.
Testing Option (ii):
A=(1001−12),v=1−2−1Av=(1⋅1+0⋅(−2)+(−1)⋅(−1)0⋅1+1⋅(−2)+2⋅(−1))=(1+1−2−2)=(2−4)
This is incorrect.
Testing Option (iii):
A=(100112),v=1−2−1Av=(1⋅1+0⋅(−2)+1⋅(−1)0⋅1+1⋅(−2)+2⋅(−1))=(1−1−2−2)=(0−4)
This is incorrect.
Conclusion:
The correct matrix for A is Option (i): (10011−2).
Would you like further details on any of the steps? Here are some related questions:
How is the row-reduced echelon form (RREF) of a matrix determined?
What is the geometric interpretation of the solution to a homogeneous system Av=0?
How do we compute matrix-vector multiplication?
Why is the vector v called a solution to Av=0?
What are the key properties of matrices in RREF?
Tip: Always check if a matrix is in row-reduced echelon form by verifying if each leading entry is 1, and all entries above and below the leading entry in the column are zeros.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra Matrix-Vector Multiplication RREF (Row-Reduced Echelon Form) Homogeneous Systems
Formulas
Matrix-vector multiplication RREF properties
Theorems
Properties of RREF Solution to homogeneous systems