Math 225 Exam -2 questions with answers

Math 225 Exam #2 questions with |\ |\ |\ |\ |\ |\




answers


Invertible Matrix Theorem - CORRECT ANSWERS ✔✔Let A be a
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square n x n matrix, then the following statements are either all
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true or all false...
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-A is an invertible matrix
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-A is row equivalent to the n x n identity matrix
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-A has n pivot positions
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-The equation Ax=0 has only the trivial solution
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-The columns of A form a linearly independent set
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-The SLE Ax=b has a solution for each b in Rn
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-The columns of A span Rn
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-There is an n x n matrix C such that CA=I
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-There is an n x n matrix D such that AD=I
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-A^T is an invertible matrix
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Determinant - CORRECT ANSWERS ✔✔det A= a11detA11 - |\ |\ |\ |\ |\ |\ |\ |\


a12detA12 + ... + (-1)^(1+n)a1ndetA1n |\ |\ |\ |\




The determinant of an n x n matrix A can be computed by... -
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CORRECT ANSWERS ✔✔... a cofactor expansion across any row or
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down any column
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, Cofactor expansion across the ith row - CORRECT ANSWERS
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✔✔det A= ai1Ci1 + ai2Ci2 + ... + ainCin
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=(-1)^(i+1)*ai1*detAi1 + (-1)^(i+2)*ai2*detAi2+ ... + (- |\ |\ |\ |\ |\


1)^(i+n)*ain*detAin


Cofactor expansion down jth column - CORRECT ANSWERS ✔✔det
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A= a1jC1j + a2jC2j + ... + anjCnj
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=(-1)^(1+j)*a1j*detA1j + (-1)^(2+j)*a2j*detA2j+ ... + (- |\ |\ |\ |\ |\


1)^(n+j)*anj*detAnj


If A is a triangular matrix,... - CORRECT ANSWERS ✔✔... then det
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A is the product of the entries on the main diagonal of A
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Effects of row operations on determinants - CORRECT ANSWERS
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✔✔Let A be a square matrix |\ |\ |\ |\ |\




-If a multiple of one row of A is added to another row to produce
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a matrix B, then det B= det A
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-If two rows are interchanged (swapped) to produce B, then det
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B= -det A |\ |\




-If one row of A is multiplied by k to produce B, then det B =
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k*det A |\




det = (-1)^r*det U... - CORRECT ANSWERS ✔✔... when A is
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invertible


det = 0 - CORRECT ANSWERS ✔✔... when A is not invertible
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