Explanation: Symmetry Aᵀ=A is defined only for square matrices because aᵢⱼ must match aⱼᵢ for every pair. That requires same number of rows and columns (n×n).
For matrices A and B, if AB is defined, the number of columns of AB equals
A Rows of A
B Rows of B
C Columns of B
D Columns of A
Explanation: If A is m×n and B is n×p, then AB is m×p. The product keeps rows from A and columns from B, so columns of AB equal columns of B.
When is BA defined if A is 2×3 and B is 3×2
A Never defined
B Always defined
C Only if square
D Only if det nonzero
Explanation: B is 3×2 and A is 2×3, so BA is (3×2)(2×3) which is defined because inner dimensions match (2). Result will be 3×3.
If A is 2×3 and B is 3×2, order of BA is
A 2×2
B 2×3
C 3×2
D 3×3
Explanation: BA multiplies B(3×2) with A(2×3). Inner dimensions 2 match, so output has outer dimensions 3×3. This also shows AB and BA can have different orders.
For any square matrix A, det(−A) for n×n equals
A (−1)ⁿ det(A)
B −det(A) always
C det(A)/2
D det(A)²
Explanation: det(kA)=kⁿ det(A) for n×n matrices. Taking k=−1 gives det(−A)=(−1)ⁿ det(A). Sign depends on whether n is even or odd.
If A is 4×4 and det(A)=3, then det(−A) equals
A −3
B 3
C 12
D 1/3
Explanation: Here n=4, so (−1)⁴=+1. Therefore det(−A)=det(A)=3. For even order matrices, multiplying by −1 does not change determinant sign.
If A is 3×3 and det(A)=3, then det(−A) equals
A 3
B 9
C 1/3
D −3
Explanation: For n=3, (−1)³=−1. So det(−A)=−det(A)=−3. For odd order matrices, multiplying matrix by −1 flips determinant sign.
If A is invertible, then det(AᵀA) equals
A det(A)
B 1/det(A)
C det(A)²
D 0 always
Explanation: det(AᵀA)=det(Aᵀ)det(A)=det(A)det(A)=det(A)², since det(Aᵀ)=det(A). This is always positive if det(A) is real.
If A is orthogonal, then det(A) must be
A 0 only
B ±1
C 1 only
D Any real
Explanation: Orthogonal means AᵀA=I. Taking determinants gives det(Aᵀ)det(A)=det(I)=1, so det(A)²=1 and det(A)=±1 for real matrices.
For a diagonal matrix, determinant equals
A Sum of diagonal
B Sum of all entries
C Always 1
D Product diagonal
Explanation: A diagonal matrix is triangular, so its determinant equals the product of diagonal entries. This holds for any upper/lower triangular matrix, not only diagonal.
If A has a row of all zeros, then det(A) is
A 0
B 1
C Depends on trace
D Depends on rank
Explanation: If any row (or column) is entirely zero, rows are linearly dependent, so determinant becomes zero. Such a matrix is singular and cannot have an inverse.
If A has two proportional rows, then det(A) is
A Nonzero
B Always 1
C Zero
D Always −1
Explanation: Proportional rows mean linear dependence. Determinant measures independence of rows/columns; dependence forces det(A)=0, so the matrix is singular and inverse does not exist.
If A is invertible, then (A⁻¹)ᵀ equals
A (Aᵀ)⁻¹
B (Aᵀ)⁻¹
C −A⁻¹
D adj(A)
Explanation: For invertible A, transpose and inverse commute in this way: (Aᵀ)⁻¹=(A⁻¹)ᵀ. It follows by transposing AA⁻¹=I and using uniqueness of inverse.
If AB=I, then B is
A adj(A)
B Aᵀ
C Zero matrix
D A⁻¹
Explanation: If AB=I for square matrices, B is a right-inverse of A. For square matrices, right-inverse equals inverse, so B must be A⁻¹ and also BA=I.
If A is invertible, then inverse of A⁻¹ is
A A
B Aᵀ
C adj(A)
D 0 matrix
Explanation: Inverse reverses multiplication. Since A⁻¹ is defined by AA⁻¹=I, taking inverse of both sides gives (A⁻¹)⁻¹=A. So the inverse of inverse returns original.
If det(A)=2, then det(adj(A)) for 3×3 A equals
A 2
B 4
C 8
D 16
Explanation: For n×n matrix, det(adj(A)) = (det(A))^(n−1). Here n=3, so det(adj(A))=(2)^(2)=4. This holds for any square matrix.
For n×n matrix, A·adj(A) equals
A det(A)A
B adj(A)I
C det(A)I
D det(A)0
Explanation: A·adj(A)=det(A)I is a standard identity. It connects determinant and adjoint and is the basis for inverse formula A⁻¹=adj(A)/det(A) when det(A)≠0.
If det(A)=0, then A·adj(A) equals
A I
B det(A)I
C 0 matrix
D adj(A)
Explanation: The identity A·adj(A)=det(A)I holds for all square matrices. If det(A)=0, then det(A)I becomes the zero matrix, so product equals zero matrix.
A matrix satisfying A²=I is best called
A Idempotent
B Nilpotent
C Singular
D Involutory
Explanation: Involutory matrices satisfy A²=I, meaning applying the transformation twice returns the original. Such matrices are self-inverse: A⁻¹=A, provided they are square.
A matrix satisfying A²=A is best called
A Involutory
B Orthogonal
C Idempotent
D Skew-symmetric
Explanation: Idempotent means A²=A. These matrices often represent projection operations in linear algebra. They are not necessarily invertible unless A=I.
For any square A, trace(Aᵀ) equals
A −trace(A)
B trace(A)
C 1/trace(A)
D trace(A)²
Explanation: Trace is sum of diagonal entries. Transpose does not change diagonal entries, so trace(Aᵀ)=trace(A). This is a basic and useful property in matrix algebra.
Trace is linear, meaning trace(A+B) equals
A trace(A)trace(B)
B trace(A)−trace(B)
C trace(A)/trace(B)
D trace(A)+trace(B)
Explanation: Trace is linear: trace(A+B)=trace(A)+trace(B) and trace(kA)=k·trace(A). This holds because diagonal entries add directly under matrix addition.
For square matrices, trace(AB) equals
A trace(A)+trace(B)
B trace(A)trace(B)
C trace(BA)
D trace(AᵀBᵀ) only
Explanation: For compatible square matrices, trace(AB)=trace(BA). This cyclic property helps simplify expressions, though in general AB≠BA. Trace cares about diagonal sums, not full equality.
If A is skew-symmetric of odd order, det(A) is
A Always 0
B Always 1
C Always −1
D Any real
Explanation: For skew-symmetric A, det(A)=det(Aᵀ)=det(−A)=(−1)ⁿ det(A). If n is odd, this implies det(A)=−det(A), so det(A)=0.
If a matrix has det(A)=0, then system AX=B has
A Always unique
B Always solvable
C Always inconsistent
D Not guaranteed unique
Explanation: det(A)=0 means A is singular, so inverse method fails. The system may have no solution or infinitely many solutions, depending on whether it is consistent.
For a homogeneous system AX=0 with det(A)≠0, solutions are
A Infinitely many
B Only trivial
C No solution
D Two solutions
Explanation: If det(A)≠0, A is invertible. Multiply AX=0 by A⁻¹ to get X=0. So the homogeneous system has only the trivial solution.
For a homogeneous system AX=0 with det(A)=0, it has
A Only trivial
B No solution
C At least one nontrivial
D Unique always
Explanation: If det(A)=0, A is singular, so rank(A)
Cramer’s rule is valid for
A Square with det≠0
B Rectangular systems
C Any consistent system
D Only homogeneous system
Explanation: Cramer’s rule requires a square coefficient matrix with nonzero determinant. Then each variable is a ratio of determinants, producing a unique solution.
In Cramer’s rule, Dᵢ is formed by replacing
A i-th row
B main diagonal
C last column only
D i-th column
Explanation: For AX=B, D is det(A). Dᵢ is determinant of matrix formed by replacing the i-th column of A with the constant column B, keeping all other columns same.
If det(A)=0 and rank(A)=rank([A|B])=r<n, system has
A No solution
B Infinite solutions
C Unique solution
D Two solutions
Explanation: Equal ranks mean consistency. If rank is less than number of variables, at least one free variable exists. That gives infinitely many solutions, typically written in parametric form.
If rank(A) < rank([A|B]), then the system is
A Consistent
B Unique always
C Homogeneous
D Inconsistent
Explanation: When augmented matrix rank is larger, elimination produces a contradiction like 0=nonzero. That means no solution exists, so the system is inconsistent.
In row echelon form, pivot positions are located
A Any column
B Always last column
C Moving right downward
D Only diagonal
Explanation: In echelon form, each leading nonzero entry (pivot) of a lower row appears to the right of the pivot in the row above. This structure helps count rank and solve.
In reduced row echelon form, each pivot column has
A Zeros above and below
B Zeros below only
C Ones only
D Random entries
Explanation: In RREF, each pivot is 1 and is the only nonzero entry in its column. That means all entries above and below pivot become zero, simplifying solution reading.
Determinant changes sign when performing
A Rᵢ → Rᵢ + kRⱼ
B Rᵢ ↔ Rⱼ
C Multiply whole matrix
D Add columns together
Explanation: Swapping two rows interchanges orientation and flips determinant sign. Other operation like adding a multiple of another row does not change determinant value.
If one row is multiplied by 3 and another by 2, det scales by
A 5
B 1/6
C 0
D 6
Explanation: Determinant is linear in each row. Scaling one row by 3 multiplies det by 3, scaling another row by 2 multiplies by 2 more, giving total factor 3×2=6.
For 3×3 A, if two rows are swapped twice, determinant becomes
A −D
B 0
C D
D 2D
Explanation: Each row swap multiplies determinant by −1. Two swaps multiply by (−1)²=+1, so determinant returns to original value D. Multiple swaps follow sign parity.
If det(A)=5, det(Aᵀ)=
A 0
B 5
C −5
D 25
Explanation: Determinant does not change under transpose: det(Aᵀ)=det(A). So if det(A)=5, determinant of transpose is also 5.
If A is 3×3 and det(A)=2, then det(A²) equals
A 2
B 6
C 8
D 4
Explanation: det(A²)=det(A·A)=det(A)det(A)=(det(A))². So with det(A)=2, det(A²)=4. This holds for any square matrix.
If A is invertible, then det(Aᵀ)det(A⁻¹) equals
A 0
B 1
C det(A)
D det(A)²
Explanation: det(Aᵀ)=det(A) and det(A⁻¹)=1/det(A). Multiplying gives det(A)·(1/det(A))=1, confirming determinant relationships for inverse and transpose.
If A is 2×2 with det(A)=1, then det(adj(A)) equals
A 0
B 2
C 1
D −1
Explanation: For n×n, det(adj(A))=(det(A))^(n−1). For 2×2, power is 1, so det(adj(A))=det(A)=1. This matches adj(A)=det(A)A⁻¹.
Which type of matrix commutes with every square matrix B of same order
A Diagonal matrix
B Symmetric matrix
C Triangular matrix
D Scalar matrix
Explanation: A scalar matrix has the form kI. For any compatible matrix B, (kI)B = kB and B(kI) = kB, so they are always equal. General diagonal/symmetric/triangular matrices do not always commute.
A matrix P satisfying P²=P and Pᵀ=P is called
A Projection matrix
B Rotation matrix
C Nilpotent matrix
D Singular always
Explanation: P²=P makes it idempotent and Pᵀ=P makes it symmetric. Such matrices represent orthogonal projections onto a subspace in linear algebra.
If A is orthogonal, then (A⁻¹)ᵀ equals
A Aᵀ
B A
C −A
D adj(A)
Explanation: For orthogonal A, A⁻¹=Aᵀ. Then (A⁻¹)ᵀ=(Aᵀ)ᵀ=A. So transpose of inverse returns the original orthogonal matrix.
If A is invertible and AB=AC, then B equals
A A
B B+C
C C
D Cannot decide
Explanation: If AB=AC and A is invertible, multiply both sides by A⁻¹ on left: A⁻¹AB=A⁻¹AC, giving IB=IC, so B=C. Invertibility allows cancellation.
If A is invertible and BA=CA, then B equals
A A
B I
C Cannot decide
D C
Explanation: Since A is invertible, multiply BA=CA on the right by A⁻¹: BAA⁻¹=CAA⁻¹, giving BI=CI, so B=C. Right multiplication works similarly.
A 2D area by determinant of vectors u=(x₁,y₁), v=(x₂,y₂) is
A x₁x₂+y₁y₂
B |x₁y₂−y₁x₂|
C x₁y₂+y₁x₂
D |x₁x₂−y₁y₂|
Explanation: The area of the parallelogram formed by two vectors in 2D equals absolute determinant |x₁ y₁; x₂ y₂| = |x₁y₂−y₁x₂|. Triangle area is half of it.
The 2D triangle area from two vectors u and v is
A |det(u,v)|
B 2|det(u,v)|
C (1/2)|det(u,v)|
D det(u,v) only
Explanation: Parallelogram area formed by u and v is |det(u,v)|. A triangle with same vectors as sides has half that area, so triangle area is (1/2)|det(u,v)|.
In simple cryptography, a message vector is often transformed by
A Matrix multiplication
B Transpose only
C Determinant expansion
D Row swapping
Explanation: Many basic encryption schemes represent letters as numbers and multiply message vectors by a key matrix. Decoding requires the inverse key matrix, so determinant must be nonzero.
A matrix with det(A)=1 preserves
A Area only
B Volume scale
C Orientation always
D Length always
Explanation: Determinant gives scaling of area/volume under a linear transformation. If det(A)=1, it preserves area in 2D and volume in 3D (scale factor 1), though shapes may change.
If det(A)=−1, then the transformation typically
A Preserves orientation
B Makes zero area
C Reverses orientation
D Becomes singular
Explanation: Negative determinant indicates orientation reversal (like reflection), while |det(A)| gives scale factor. det(A)=−1 keeps area/volume magnitude the same but flips orientation direction