Echelon Exam
2.1 (a) The determinant of the 2×2 matrix
[a b; c d] is ad+bc. - ✔ ✔ False
2.1 (b) Two square matrices that have the same determinant must
have the same size. - ✔ ✔ False
2.1 (c) The minor Mij is the same as the cofactor Cij if i+j is even -
✔ ✔ True
2.1 (d) If A is a 3×3 symmetric matrix, then Cij =Cji for all i and j. -
✔ ✔ True
2.1 (e) The number obtained by a cofactor expansion of a matrix
A is independent of the row or column chosen for the expansion. -
✔ ✔ True
2.1 (f) If A is a square matrix whose minors are all zero, then
det(A)=0. - ✔ ✔ True
2.1 (g) The determinant of a lower triangular matrix is the sum of
the entries along the main diagonal. - ✔ ✔ False
2.1 (h) For every square matrix A and every scalar c, it is true that
det(cA)=cdet(A). - ✔ ✔ False
2.1 (i) For all square matrices A and B, it is true that:
det(A+B)=det(A)+det(B) - ✔ ✔ False
2.1 (j) For every 2×2 matrix A it is true that det(A^2)=(det(A))^2. -
✔ ✔ True
, 2.2 (a) If A is a 4×4 matrix and B is obtained from A by
interchanging the first two rows and then interchanging the last
two rows, then det(B)=det(A). - ✔ ✔ True
2.2 (b) If A is a 3×3 matrix and B is obtained from A by multiplying
the first column by 4 and multiplying the third column by 3/4, then
det(B)=3det(A). - ✔ ✔ True
2.2 (c) If A is a 3×3 matrix and B is obtained from A by adding 5
times the first row to each of the second and third rows, then
det(B)=25det(A). - ✔ ✔ False
2.2 (d) If A is an n×n matrix and B is obtained from A by
multiplying each row of A by its row number, then:
det(B) = (n(n+1)/2)det(A) - ✔ ✔ False
2.2 (e) If A is a square matrix with two identical columns, then
det(A)=0. - ✔ ✔ True
2.2 (f) If the sum of the second and fourth row vectors of a 6×6
matrix A is equal to the last row vector, then det(A)=0. - ✔ ✔
True
2.3 (a) If A is a 3×3 matrix, then det(2A)=2det(A). - ✔ ✔ False
2.3 (b) If A and B are square matrices of the same size such that
det(A)=det(B), then det(A+B)=2det(A). - ✔ ✔ False
2.3 (c) If A and B are square matrices of the same size and A is
invertible, then:
det(A^(-1)BA) = det(B) - ✔ ✔ True
2.1 (a) The determinant of the 2×2 matrix
[a b; c d] is ad+bc. - ✔ ✔ False
2.1 (b) Two square matrices that have the same determinant must
have the same size. - ✔ ✔ False
2.1 (c) The minor Mij is the same as the cofactor Cij if i+j is even -
✔ ✔ True
2.1 (d) If A is a 3×3 symmetric matrix, then Cij =Cji for all i and j. -
✔ ✔ True
2.1 (e) The number obtained by a cofactor expansion of a matrix
A is independent of the row or column chosen for the expansion. -
✔ ✔ True
2.1 (f) If A is a square matrix whose minors are all zero, then
det(A)=0. - ✔ ✔ True
2.1 (g) The determinant of a lower triangular matrix is the sum of
the entries along the main diagonal. - ✔ ✔ False
2.1 (h) For every square matrix A and every scalar c, it is true that
det(cA)=cdet(A). - ✔ ✔ False
2.1 (i) For all square matrices A and B, it is true that:
det(A+B)=det(A)+det(B) - ✔ ✔ False
2.1 (j) For every 2×2 matrix A it is true that det(A^2)=(det(A))^2. -
✔ ✔ True
, 2.2 (a) If A is a 4×4 matrix and B is obtained from A by
interchanging the first two rows and then interchanging the last
two rows, then det(B)=det(A). - ✔ ✔ True
2.2 (b) If A is a 3×3 matrix and B is obtained from A by multiplying
the first column by 4 and multiplying the third column by 3/4, then
det(B)=3det(A). - ✔ ✔ True
2.2 (c) If A is a 3×3 matrix and B is obtained from A by adding 5
times the first row to each of the second and third rows, then
det(B)=25det(A). - ✔ ✔ False
2.2 (d) If A is an n×n matrix and B is obtained from A by
multiplying each row of A by its row number, then:
det(B) = (n(n+1)/2)det(A) - ✔ ✔ False
2.2 (e) If A is a square matrix with two identical columns, then
det(A)=0. - ✔ ✔ True
2.2 (f) If the sum of the second and fourth row vectors of a 6×6
matrix A is equal to the last row vector, then det(A)=0. - ✔ ✔
True
2.3 (a) If A is a 3×3 matrix, then det(2A)=2det(A). - ✔ ✔ False
2.3 (b) If A and B are square matrices of the same size such that
det(A)=det(B), then det(A+B)=2det(A). - ✔ ✔ False
2.3 (c) If A and B are square matrices of the same size and A is
invertible, then:
det(A^(-1)BA) = det(B) - ✔ ✔ True
