LINEAR ALGEBRA EXAM 2 QUESTIONS
AND ANSWERS 2024/2025
define a cofactor for each entry based on its minor - Answer-Let A be an nxn matrix,
with n>=2. The (i,j) cofactor of A, Aij is (-1)^(i+j) times the (i,j) minor of A. That is Aij = (-
1)^(i+j) |Aij|
determinant - Answer-Let A be an nxn (square matrix). The deterimant of A, denoted |
A|, is defined as follows: If n=1 (so that A=[a11]), then |A|=a11. If n>1, then |A|
=an1An1+an2An2+...+annAnn
Theorem 3.2: Determinants of Upper Triangular Matrices - Answer-Let A be an upper
triangular nxn matrix. Then |A|=a11*a22*...*ann, the product of the entries of A along
the main diagonal
Proof of theorem 3.2 - Answer-Induction on n. Let n=1. In this case, A=[a11], and |A|
=a11, which verifies the formula in the theorem. Let n>1. Assume that for any upper
triangular (n-1)x(n-1) matrix B, |B|=b11*b22*...*b(n-1)(n-1). We must prove that the
formula given in the theorem holds for any nxn matrix A. Now, |A|=an1An1+an2An2+...
+annAnn=0An1+0n2+...+0A(n-1)(n-1)+annAnn, because ani=0 for i<n since A is upper
triangular. Thus, |A|=annAnn=ann(-1)^(n+n)|Ann|=ann|Ann| (since n is even). However,
the (n-1)x(n-1) submatrix Ann is itself an upper triangular matrix, since A is upper
triangular. Thus by the inductive hypothesis, |Ann|=a11*a22*...*a(n-1)(n-1). Hence, |A|
=ann(a11*a22*...*a(n-1)(n-1))=a11a22...ann.
Theorem 3.3: Effect of Row operations on the Determinant - Answer-Let A be an nxn
matrix, with determinant |A|, and let c be a scalar.
1) if R1 is the Type(I) row operation <i>-->c<i>, then |R1(A)|=c|A|
2) If R2 is the Type (II) row operation <j>-->c<i>+<j>, then |R2(A)|=|A|
3) If R3 is the Type (III) row operation <i><--><j>, then |R3A|=-|A|
Corallary 3.4: If A is n × n, then |cA| = cn|A|. - Answer-Part (1) of Theorem 3.3 can be
used to multiply each of the n rows of a matrix A by c in turn, thus proving the following
corollary: If A is n × n, then |cA| = cn|A|.
Theorem 3.5: An nxn matrix A is nonsingular if an only if |A| does not equal 0 - Answer-
Let D be the unique matrix in reduced row echelon form A. Now, using theorem 3.3 we
see that a single row operation of any type cannot convert a matrix having a nonzero
determinant to have have a zero determinant. Because A is converted to D using a finite
number of such row operations, Theorem 3.3 assures us that |A| and |D| are either both
zero or nonzero. Now, if A is nonsingular (Which implies D=In), we know that |D|=1 and
therefore |A| cannot equal 0. Assume |A| does not equal zero. Then we know that |D|
does not equal 0. Because D is a square matrix with a staircase pattern of pivots, it is
upper triangular. Because |D| does not equal 0, Theorem 3.2 asserts that all main
AND ANSWERS 2024/2025
define a cofactor for each entry based on its minor - Answer-Let A be an nxn matrix,
with n>=2. The (i,j) cofactor of A, Aij is (-1)^(i+j) times the (i,j) minor of A. That is Aij = (-
1)^(i+j) |Aij|
determinant - Answer-Let A be an nxn (square matrix). The deterimant of A, denoted |
A|, is defined as follows: If n=1 (so that A=[a11]), then |A|=a11. If n>1, then |A|
=an1An1+an2An2+...+annAnn
Theorem 3.2: Determinants of Upper Triangular Matrices - Answer-Let A be an upper
triangular nxn matrix. Then |A|=a11*a22*...*ann, the product of the entries of A along
the main diagonal
Proof of theorem 3.2 - Answer-Induction on n. Let n=1. In this case, A=[a11], and |A|
=a11, which verifies the formula in the theorem. Let n>1. Assume that for any upper
triangular (n-1)x(n-1) matrix B, |B|=b11*b22*...*b(n-1)(n-1). We must prove that the
formula given in the theorem holds for any nxn matrix A. Now, |A|=an1An1+an2An2+...
+annAnn=0An1+0n2+...+0A(n-1)(n-1)+annAnn, because ani=0 for i<n since A is upper
triangular. Thus, |A|=annAnn=ann(-1)^(n+n)|Ann|=ann|Ann| (since n is even). However,
the (n-1)x(n-1) submatrix Ann is itself an upper triangular matrix, since A is upper
triangular. Thus by the inductive hypothesis, |Ann|=a11*a22*...*a(n-1)(n-1). Hence, |A|
=ann(a11*a22*...*a(n-1)(n-1))=a11a22...ann.
Theorem 3.3: Effect of Row operations on the Determinant - Answer-Let A be an nxn
matrix, with determinant |A|, and let c be a scalar.
1) if R1 is the Type(I) row operation <i>-->c<i>, then |R1(A)|=c|A|
2) If R2 is the Type (II) row operation <j>-->c<i>+<j>, then |R2(A)|=|A|
3) If R3 is the Type (III) row operation <i><--><j>, then |R3A|=-|A|
Corallary 3.4: If A is n × n, then |cA| = cn|A|. - Answer-Part (1) of Theorem 3.3 can be
used to multiply each of the n rows of a matrix A by c in turn, thus proving the following
corollary: If A is n × n, then |cA| = cn|A|.
Theorem 3.5: An nxn matrix A is nonsingular if an only if |A| does not equal 0 - Answer-
Let D be the unique matrix in reduced row echelon form A. Now, using theorem 3.3 we
see that a single row operation of any type cannot convert a matrix having a nonzero
determinant to have have a zero determinant. Because A is converted to D using a finite
number of such row operations, Theorem 3.3 assures us that |A| and |D| are either both
zero or nonzero. Now, if A is nonsingular (Which implies D=In), we know that |D|=1 and
therefore |A| cannot equal 0. Assume |A| does not equal zero. Then we know that |D|
does not equal 0. Because D is a square matrix with a staircase pattern of pivots, it is
upper triangular. Because |D| does not equal 0, Theorem 3.2 asserts that all main
