1. [10 pts] Determine if the following statements are true or false. Explain why
ID: 3114152 • Letter: 1
Question
1. [10 pts] Determine if the following statements are true or false. Explain why and circle your answer. Only cireling "true" or "false" will receive little credit. (a) If the number of the equations in a linear system exceeds the number of unknowns, then the system must be inconsistent. True False (b) If A and B are square matrices of the same size, then det(A+B)-det( A)+det(B). True False (c) If A is invertible and a multiple of the first row of A is added to the third row, then the resulting matrix must be invertible. True False (d) If nxn matrix A is not invertible, then the homogeneous system AX-0 has infinitely many solutions. True False (e) If A and B are matrices of same size and k is a constant, then (kA+ B kA+B True FalseExplanation / Answer
a)False
eg ; conside the equations x+y=4,x-y=2,2x+2y=8
no of equations are 3 and no of unknowns are 2 , but has consistant solutions x=3,y=2
b)False
eg:let B=A, A be nxn matrix
det(2A)=2ndet(A) != 2det(A)
c)True
As A is a invertible matrix det(A)!=0
Performimg column operations on matrix dosen't effect its det
So the det will be non 0 after performing column operations
which implies it is invertable
d)True
det is zero which implies atlest one row can be written in the forn of other rows
there are n distinguish variables and less than n rows ,So there will be infinite solutions
e)True
Transpose satisfies Associtivity
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