Elementary Linear Algebra
If A has a row of zeros, so also does adj A
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If AB = AC and A doesn't equal 0, then B = C.
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If AB has a column of zeros, so also does B.
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If AB has a row of zeros, so also does A.
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If AB is invertible, then A and B are invertible
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If A^T = −A, then det A = −1
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If Ax=b has a solution for some columb b then it has a solution for every column b
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If R is the reduced row-echelon form of A, then det A = det R
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If U 6= R n is a subspace of R n and x + y is in U, then x and y are both in U
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If adj A = 0, then A = 0
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If adj A exists, then A is invertible
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If all of x1, x2, ..., xk are nonzero, then {x1, x2, ..., xk} is independent
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If ax+by+cz = 0, then {x, y, z} is independent
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If det A = 0, then A has two equal rows
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If det A = 1, then adj A = A
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If det A = det B where A and B are the same size, then A = B
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If {x, y} is independent, then {x, y, x+ y} is independent
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The empty set of vectors in R^n is a subspace of R^n
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[0;1] is in span {[1;0] , [2;0]}
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a consistent linear system must have infinitely many solutions
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det(A+B) = det A+det B
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det(AB) = det(B^TA)
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det(A^T) = − det A
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det(A^TA) > 0 for all square matrices A
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det(I +A) = 1+ det A
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det(−A) = − det A
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if A = [a1 a2 a3] in terms of its columns and if the system Ax=b has a solution, then b=sa1+ta2 for some s,t
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if A and B are both invertible then A+B is invertible
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if A doesn't equal 0 is a square matrix then A is invertible
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if A doesn't equal 0 then A^2 doesn't equal 0
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if A has a row of zeros so also does BA for all B
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if A has a row of zeros, there is more than one solution
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if A is mxn and m<n then Ax=b has a solution for every column b
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if A+B=0 then B=0
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if AB=B for some B doesn't equal 0 then A is invertible
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if AB=I then A and B commute
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if AJ=A then J=I
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if A^2=A and A doesn't equal 0 then A is invertible
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if A^2=I then A=I
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if Ax has a zero entry then A has a row of zeros
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if Ax=0 where x doesn't equal 0 then A=0
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if a linear system has n variables and m equations, then the augmented matrix has n rows
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if the (3,1) entry of A is 5 then (1,3) entry of A^T is -5
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if the rank A=3 the system is consistent
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if the row echelon form of A has a row of zeros there exist nontrivial solutions
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if the row echelon form of C has a row of zeros, there is no solution
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if the system has a nontrivial solution, it cannot be homogeneous
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if the system is consistent for some choice of constants, it is consistent for every choice of constants
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if the system is consistent, it must be homogeneous
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if the system is homogeneous every solution is trivial
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if there exist nontrivial solutions, the row echelon form of A has a row of zeroes
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if there exists a nontrivial solution, there is no trivial solution
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if there exists a solution, there are infinitely many solutions
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if there is more than one solution A has a row of zeros
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A and A^T have the same main diagonal for every matrix A
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If A commutes with A+B then A commutes with B
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If A has a row of zeros, so also does AB.
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If A is 2×2, then det(7A) = 49 det A.
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If A is 2×2, then det(A^T) = det A.
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If A is invertible and adj A = A^−1 , then det A = 1
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If A is invertible, then adj A is invertible
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If A is symmetric, then I +A is symmetric.
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If B has a column of zeros so also does AB
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If U is a subspace of R n and rx is in U for all r in R, then x is in U
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If U is a subspace of R n and x is in U, then −x is also in U
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If det A 6= 0 and AB = AC, then B = C
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If one of x1, x2, ..., xk is zero, then {x1, x2, ..., xk} is dependent
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If x is in U and U = span {y, z}, then U = span {x, y, z}
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If {x, y, z} is independent, then {y, z} is independent
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If {y, z} is dependent, then {x, y, z} is dependent for any x
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[3;2] is a linear combination of [1;0] and [0;1]
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every linear combination of vectors in R^n can be written in the form Ax
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if A = [a1 a2 a3] in terms of its columns, and b = 3a1-2a2 then the system Ax=b has a solution
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if A and B are both invertible then (A^-1B)^T is invertible
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if A and B are symmetric, then kA and mB is symmetric for any scalars k and m
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if A is invertible and skew symmetric (A^T = -A) the same is true of A^-1
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if A is square, then (A^T)^3 = (A^3)^T
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if A+B=A+C then B and C have the same size
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if A^2 is invertible then A is invertible
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if A^4=3I then A is invertible
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if B is symmetric and A^T=3B then A=3B
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if a row operation is applied to the system, the new system is also homogeneous
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if a row operation is done to a consistent linear system, the resulting system must be consistent.
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if a series of row operations on a linear system results in an inconsistent system, the original system is inconsistent
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if the rank C=3 the system is consistent
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if the system is consistent, there is more than one solution
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if there exists a trivial solution, the system is homogeneous
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if there is no solution, the reduced row echelon form of C has a row of zeros
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if x1 and x2 are solutions to Ax=b then x1-x2 is a solution to Ax=0
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let A = [a1 a2 a3] in terms of it columns. if a3= sa1 + ta2 then ax=0 where x = [s;t;-1]
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m. If A is invertible and det A = d, then adj A = dA^−1
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the rank of A is at most 3
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there is no system that is inconsistent for every choice of constants
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1 0 -1 2 / 1 0 0 1 4 / 1 0 0 0 1 / 3
infinitely many solutions
1 0 0 / 1 0 1 0 / 1 0 0 1 / 1 1 1 1 / 6
no solution
1 1 1 2 / 1 0 0 1 5 / 1 0 0 0 0 /3
no solution
0 0 2 \ 1 0 1 4 \ 2 1 -1 2 \ 3
unique solution
1 0 0 / 5 1 1 0 / 1 2 -1 2 / 1
unique solution
1 0 0 0 / 0 0 4 0 0 / 0 0 0 2 0 / 0 0 0 0 1 / 0
unique solution