Chapter 2 Linear Algebra Test
Assume 𝐴 is an 𝑛×𝑛 matrix. Answer "True" if the statement is always true, and "False" otherwise. If det 𝐴 is zero, then two columns of 𝐴 must be the same, or all of the elements in a row or column of 𝐴 are zero.
False
Assume 𝐴 is an 𝑛×𝑛 matrix. Answer "True" if the statement is always true, and "False" otherwise. If the columns of 𝐴 are linearly independent, then det 𝐴=0
False
Assume 𝐴 is an 𝑛×𝑛 matrix. Answer "True" if the statement is always true, and "False" otherwise. det(𝐴+𝐵)=det(𝐴)+det(𝐵)
False
Suppose 𝐴 and 𝐵 are invertible matrices. Mark each statement as true or false. True means that the statement is true for all invertible matrices 𝐴 and 𝐵. (𝐴+𝐵)2=𝐴2+𝐵2+2𝐴𝐵
False
Suppose 𝐴 and 𝐵 are invertible matrices. Mark each statement as true or false. True means that the statement is true for all invertible matrices 𝐴 and 𝐵. (𝐴𝐵)−1=𝐴^−1𝐵^−1
False
Suppose 𝐴 and 𝐵 are invertible matrices. Mark each statement as true or false. True means that the statement is true for all invertible matrices 𝐴 and 𝐵. 𝐴+𝐵 is invertible.
False
Suppose 𝐴 is an 𝑛×𝑛 matrix. Select "yes" if the statement is always true, "no" if the statement is always false, and "maybe" if the statement is sometimes true and sometimes false. If the equation 𝐴𝑥=0 has the trivial solution, then the columns of 𝐴 span 𝐑𝑛.
Maybe
Suppose 𝐴 is an 𝑛×𝑛 matrix. Select "yes" if the statement is always true, "no" if the statement is always false, and "maybe" if the statement is sometimes true and sometimes false. A square matrix with two identical columns can be invertible.
No
Suppose 𝐴 is an 𝑛×𝑛 matrix. Select "yes" if the statement is always true, "no" if the statement is always false, and "maybe" if the statement is sometimes true and sometimes false. If the linear transformation 𝑇(𝑥)=𝐴𝑥 is one-to-one, then the columns of 𝐴 form a linearly dependent set.
No
Assume 𝐴 is an 𝑛×𝑛 matrix. Answer "True" if the statement is always true, and "False" otherwise. A row replacement operation does not affect the determinant of a matrix.
True
Assume 𝐴 is an 𝑛×𝑛 matrix. Answer "True" if the statement is always true, and "False" otherwise. If two columns of 𝐴 are the same, then the determinant of that matrix is zero.
True
Suppose 𝐴 and 𝐵 are invertible matrices. Mark each statement as true or false. True means that the statement is true for all invertible matrices 𝐴 and 𝐵. (𝐼𝑛−𝐴)(𝐼𝑛+𝐴)=𝐼𝑛−𝐴^2
True
Suppose 𝐴 and 𝐵 are invertible matrices. Mark each statement as true or false. True means that the statement is true for all invertible matrices 𝐴 and 𝐵. 𝐴7 is invertible.
True
Suppose 𝐴 is an 𝑛×𝑛 matrix. Select "yes" if the statement is always true, "no" if the statement is always false, and "maybe" if the statement is sometimes true and sometimes false. If A is invertible, then the equation 𝐴𝑥=𝑏 has exactly one solution for all 𝑏 in 𝐑𝑛.
Yes
Suppose 𝐴 is an 𝑛×𝑛 matrix. Select "yes" if the statement is always true, "no" if the statement is always false, and "maybe" if the statement is sometimes true and sometimes false. If the equation 𝐴𝑥=0 has a nontrivial solution, then 𝐴 has fewer than 𝑛 pivots.
Yes
Suppose 𝐴 is an 𝑛×𝑛 matrix. Select "yes" if the statement is always true, "no" if the statement is always false, and "maybe" if the statement is sometimes true and sometimes false. If −𝐴 is not invertible, then 𝐴 is also not invertible.
Yes
Suppose 𝐴 is an 𝑛×𝑛 matrix. Select "yes" if the statement is always true, "no" if the statement is always false, and "maybe" if the statement is sometimes true and sometimes false. If 𝐴^2 is row equivalent to the n×n identity matrix, then the columns of A span 𝐑𝑛.
Yes
Suppose 𝐴 is an 𝑛×𝑛 matrix. Select "yes" if the statement is always true, "no" if the statement is always false, and "maybe" if the statement is sometimes true and sometimes false. The product of any two invertible matrices is invertible.
Yes
