Linear Algebra XM2 Exam 2026
Questions and Answers Graded A+
determinant of a 2x2 - Correct answer-ad-bc
determinant of a 3x3 - Correct answer-basketweaving method
diagonal pointed right is positive, left is negative
Theorem 3.1 Area: If x=[x1,x2] and y=[y1,y2] are nonparallel and start at the same
point in R2, then - Correct answer-the area of the parallelogram is the absolute
value of the determinant of the matrix composed of top row x and bottom row y
Theorem 3.1 Volume: If x, y, and z are vectors not on the same plane and start at
the same point in R3, then - Correct answer-the volume of the parallelogram is the
absolute value of the determinant of the matrix composed of top row x, middle row
y, and bottom row z
submatrix - Correct answer-the remaining matrix when you cross out a row and a
column of a whole matrix
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,cofactors - Correct answer-the (i, j) cofactor of A, Aij, is (-1)^i+j times the (i, j)
minor of A. The (i, j) minor of the of A is the determinant of the submatrix Aij
OR
Aij = (-1)^(i+j) |Aij|
formal definition of the determinant - Correct answer-if n>1, then
|A|=an1An1+an2An2+...+annAnn
For matrix A, each of its entry "a" on the last row is multiplied by its
corresponding cofactor. The sum of the products is the determinant
If A is an nxn matrix, prove that |cA|=c^n|A| (use induction) - Correct answer-
Show that statement holds for n=1
Assume for any kxk matrix, |cA|=c^k|A|. Then let A be a (k+1)x(k+1) matrix with
entries aij. Let B=cA and show that matrix B = c^(k+1)|A|
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,|B| = b11 |B11| + ... + b1(k+1) |B1(k+1)| *along top row
Since B=cA, bij=c(aij) and |Bij|=|cAij|, so
|B| = c(a11) |cA11| + ··· + c[a1(k+1)] |cA1(k+1)| =
c^(k+1) [a11 |A11| + ...+ a1(k+1) |A1(k+1)|]
So |B|=c^(k+1) |A| = |cA|
determinant of a Vandermonde matrix - Correct answer-1 1 1
abc
a^2 b^2 c^2
determinant = (a-b)(b-c)(c-a)
Theorem 3.2 Let A be an upper triangular n × n matrix. Then |A| = - Correct
answer-|A| = a11a22 ··· ann
the determinant is the product of the entries of A along the main diagonal
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, Theorem 3.3 Let A be an n × n matrix, with determinant |A|, and let c be a scalar.
R1, R2, and R3 are row operations such that:
R1) multiplying a row by a scalar
R2) adding a scaled row by another row
R3) swapping rows
their effects on determinant |A| is - Correct answer-|R1A| = c|A| = c^n |A| (if you
multiply a row by a scalar, multiply |A| by the reciprocal of that scalar)
|R2A| = |A|
|R3A| = -|A|
Theorem 3.5 An n × n matrix A is nonsingular iff |A| ≠ 0 - Correct answer-Let D be
the unique matrix in reduced row echelon form for A. Using Theorem 3.3, any
single row operation cannot convert a matrix with a nonzero to a matrix having a
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Questions and Answers Graded A+
determinant of a 2x2 - Correct answer-ad-bc
determinant of a 3x3 - Correct answer-basketweaving method
diagonal pointed right is positive, left is negative
Theorem 3.1 Area: If x=[x1,x2] and y=[y1,y2] are nonparallel and start at the same
point in R2, then - Correct answer-the area of the parallelogram is the absolute
value of the determinant of the matrix composed of top row x and bottom row y
Theorem 3.1 Volume: If x, y, and z are vectors not on the same plane and start at
the same point in R3, then - Correct answer-the volume of the parallelogram is the
absolute value of the determinant of the matrix composed of top row x, middle row
y, and bottom row z
submatrix - Correct answer-the remaining matrix when you cross out a row and a
column of a whole matrix
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,cofactors - Correct answer-the (i, j) cofactor of A, Aij, is (-1)^i+j times the (i, j)
minor of A. The (i, j) minor of the of A is the determinant of the submatrix Aij
OR
Aij = (-1)^(i+j) |Aij|
formal definition of the determinant - Correct answer-if n>1, then
|A|=an1An1+an2An2+...+annAnn
For matrix A, each of its entry "a" on the last row is multiplied by its
corresponding cofactor. The sum of the products is the determinant
If A is an nxn matrix, prove that |cA|=c^n|A| (use induction) - Correct answer-
Show that statement holds for n=1
Assume for any kxk matrix, |cA|=c^k|A|. Then let A be a (k+1)x(k+1) matrix with
entries aij. Let B=cA and show that matrix B = c^(k+1)|A|
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,|B| = b11 |B11| + ... + b1(k+1) |B1(k+1)| *along top row
Since B=cA, bij=c(aij) and |Bij|=|cAij|, so
|B| = c(a11) |cA11| + ··· + c[a1(k+1)] |cA1(k+1)| =
c^(k+1) [a11 |A11| + ...+ a1(k+1) |A1(k+1)|]
So |B|=c^(k+1) |A| = |cA|
determinant of a Vandermonde matrix - Correct answer-1 1 1
abc
a^2 b^2 c^2
determinant = (a-b)(b-c)(c-a)
Theorem 3.2 Let A be an upper triangular n × n matrix. Then |A| = - Correct
answer-|A| = a11a22 ··· ann
the determinant is the product of the entries of A along the main diagonal
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, Theorem 3.3 Let A be an n × n matrix, with determinant |A|, and let c be a scalar.
R1, R2, and R3 are row operations such that:
R1) multiplying a row by a scalar
R2) adding a scaled row by another row
R3) swapping rows
their effects on determinant |A| is - Correct answer-|R1A| = c|A| = c^n |A| (if you
multiply a row by a scalar, multiply |A| by the reciprocal of that scalar)
|R2A| = |A|
|R3A| = -|A|
Theorem 3.5 An n × n matrix A is nonsingular iff |A| ≠ 0 - Correct answer-Let D be
the unique matrix in reduced row echelon form for A. Using Theorem 3.3, any
single row operation cannot convert a matrix with a nonzero to a matrix having a
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