LINEAR ALGEBRA EXAM QUESTIONS AND ANSWERS

LINEAR ALGEBRA EXAM QUESTIONS
AND ANSWERS
Two matrices are equal if - Answer-they are the same size and their corresponding
entries/columns are equal

A+B=B+A - Answer-True

(A+B)+C=A+(B+C) - Answer-True

A+0=A - Answer-True

r(A+B)=rA+rB - Answer-True

(r+s)A=rA+sA - Answer-True

r(sA)=(rs)A - Answer-True

Each column of AB is a linear combination of the columns of B using weights from the
corresponding column of B.(B NOT A) - Answer-TRUE

The number of columns of A must match the number of rows in B while computing AB -
Answer-TRUE

A(BC)=(AB)C - Answer-True

A(B+C)=AB+AC - Answer-TRUE

(B+C)A=BA+BC - Answer-TRUE

R(AB)=(rA)B=A(rB) - Answer-True

(Im)A = A = A(In) - Answer-TRUE

(A^T)^T = A - Answer-TRUE

(A + B)^T = A^T + B^T. - Answer-TRUE

(rA)^T = rA^T - Answer-TRUE

(AB)^T = B^T*A^T - Answer-TRUE

Singular Matrix - Answer-a matrix that is not invertible

, Nonsingular Matrix - Answer-invertible matrix

If A is an Invertible nxn matrix, then for each b in R^n, the equation Ax=b has the unique
solution that x=A^-1 * b - Answer-TRUE, by theorem 5

(A^-1)^-1 = A - Answer-TRUE

(AB)^-1 = B^-1 * A^-1 - Answer-TRUE

(A^T)^-1 = (A^-1)^T - Answer-TRUE

The inverse of E is the elementary matrix of the same type that transforms E back into I
- Answer-TRUE

Elementary matrix E is invertible - Answer-TRUE

The range of T is the set of all linear combinations of the columns of A - Answer-TRUE

Every element row operation is reversible - Answer-TRUE

If T is a linear transformation T: R^P -> R^Q then it can represented as a... - Answer-
qxp matrix

When two linear transformations are performed after the other, the combined effect is
always a linear transformation - Answer-TRUE

If A is a 3x2 matrix the transformation x --> Ax cannot be one to one - Answer-FALSE,
linearly independent

If A is a 3x2 matrix the transformation x --> Ax cannot be onto R^3 - Answer-TRUE, not
enough pivots and columns

A linear transformation T: R^ --> R^Q is completely determined by its action on the
standard basis(the vectors) e1,e2,...ep - Answer-TRUE

If A is a 2x3 matrix the tranformation x --> Ax cannot be one to one - Answer-TRUE,
there is a free variable so NOT linearly independent

If A is a 2x3 matrix the transformation x-->Ax cannot be onto R^2 - Answer-FALSE, it is
possible since num pivots and columns match

If A, B, and C are matrices AB + AC = A(B+C) - Answer-TRUE

Given any two matrices A and B their product AB=BA - Answer-FALSE