MATRICES – Zed Vitals

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MATRICES

Test your understanding of matrices with this short quiz! Covers key concepts including matrix types, operations (addition, multiplication), determinants, inverses, and applications.

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A matrix with 3 rows and 2 columns is called a:

Square matrix

Column matrix

2 \times 3

matrix

3 \times 2

matrix

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The element $a_{23}$ in the matrix $[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}]$ is:

$a_{23}$ is the element in the 2nd row, 3rd column.

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Two matrices can be added if they have:

The same number of rows

The same number of columns

The same dimensions

The same elements

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The transpose of

$[\begin{array}{cc} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array}]$

is:

[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}]

[\begin{array}{cc} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array}]

[\begin{array}{cc} 4 & 1 \\ 5 & 2 \\ 6 & 3 \end{array}]

[\begin{array}{cc} 1 & 3 \\ 4 & 6 \end{array}]

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A matrix with only one row is called a:

Column matrix

Row matrix

Square matrix

Zero matrix

A row matrix has dimensions $1 \times n$ .

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The order of the matrix product

$A_{2 \times 3} \cdot B_{3 \times 4}$

is:

2 \times 4

3 \times 3

2 \times 3

3 \times 4

The product has rows of

$A$

and columns of

$B$

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The trace of $[\begin{array}{cc} 1 & 0 \\ 2 & 3 \end{array}]$ is:

Trace is the sum of diagonal elements: $1 + 3 = 4$ .

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A matrix is symmetric if:

A = - A

A = A^{T}

A = A^{- 1}

A

is square

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The product

$[\begin{array}{cc} 1 & 2 \end{array}] [\begin{array}{c} 3 \\ 4 \end{array}]$

is:

[\begin{array}{cc} 3 & 8 \end{array}]

[\begin{array}{c} 11 \end{array}]

[\begin{array}{c} 3 \\ 8 \end{array}]

Undefined

Dot product:

$(1) (3) + (2) (4) = 11$

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The identity matrix

$I_{3}$

is:

[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]

[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}]

[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}]

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A diagonal matrix with all diagonal elements equal to 1 is called a:

Scalar matrix

Identity matrix

Zero matrix

Symmetric matrix

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The matrix $[\begin{array}{cc} 0 & 2 \\ - 2 & 0 \end{array}]$ is:

Symmetric

Skew-symmetric

Diagonal

Identity

Skew-symmetric matrices satisfy $A^{T} = - A$ .

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A matrix with all elements zero is called a:

Identity matrix

Diagonal matrix

Zero matrix

Scalar matrix

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The matrix $[\begin{array}{cc} 5 & 0 \\ 0 & 5 \end{array}]$ is a:

Identity matrix

Scalar matrix

Zero matrix

Symmetric matrix

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An upper triangular matrix has:

Non-zero elements only above the diagonal

Non-zero elements only below the diagonal

Non-zero elements on and above the diagonal

Zero elements on the diagonal

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A matrix is orthogonal if:

A^{T} = A

A^{T} = A^{- 1}

A = - A

A

is diagonal

Orthogonal matrices satisfy

$A^{T} A = I$

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The matrix $[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}]$ is:

Diagonal

Scalar

Identity

Nilpotent

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A nilpotent matrix satisfies:

A^{T} = A

A^{k} = 0

for some

k

A^{2} = I

A

is invertible

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The matrix $[\begin{array}{ccc} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{array}]$ is:

Lower triangular

Upper triangular

Diagonal

Symmetric

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A Hermitian matrix satisfies:

A = {\overline{A}}^{T}

A = - A^{T}

A = A^{- 1}

A

is real

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$A = [\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}]$

and

$B = [\begin{array}{cc} 5 & 6 \\ 7 & 8 \end{array}]$

, then

$A + B$

is:

[\begin{array}{cc} 6 & 8 \\ 10 & 12 \end{array}]

[\begin{array}{cc} 5 & 12 \\ 21 & 32 \end{array}]

[\begin{array}{cc} 4 & 4 \\ 4 & 4 \end{array}]

[\begin{array}{cc} 6 & 12 \\ 10 & 8 \end{array}]

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The product

$[\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}] [\begin{array}{c} 5 \\ 6 \end{array}]$

is:

[\begin{array}{c} 17 \\ 39 \end{array}]

[\begin{array}{cc} 5 & 12 \\ 15 & 24 \end{array}]

[\begin{array}{c} 17 \\ 24 \end{array}]

Undefined

Row-by-column multiplication:

$(1) (5) + (2) (6) = 17$

$(3) (5) + (4) (6) = 39$

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The determinant of $[\begin{array}{cc} 2 & 3 \\ 1 & 4 \end{array}]$ is:

-5

-11

$(2) (4) - (3) (1) = 8 - 3 = 5$ .

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A matrix is invertible if its determinant is:

Zero

Non-zero

Positive

Negative

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The determinant of a $3 \times 3$ matrix $[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}]$ is:

-6

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If two rows of a matrix are identical, its determinant is:

-1

Undefined

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The determinant of $[\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array}]$ is:

-1

$(0) (0) - (1) (- 1) = 1$ .

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The determinant of an identity matrix is:

-1

Depends on size

The determinant of $I_{n}$ is always 1.

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If $A$ is $2 \times 2$ with $\det (A) = 4$ , then $\det (3 A)$ is:

For $n \times n$ matrices, $\det (k A) = k^{n} \det (A)$ .

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The determinant of a triangular matrix is:

The product of diagonal elements

The sum of diagonal elements

Zero

One

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If $\det (A) = 2$ and $\det (B) = 3$ , then $\det (A B)$ is:

-1

$\det (A B) = \det (A) \det (B)$

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The determinant of a skew-symmetric matrix of odd order is:

Zero

Positive

Negative

One

For odd $n$ , $\det (A) = 0$

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The inverse of

$[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$

is:

[\begin{matrix} 1 & 0 \\ - 1 & 1 \end{matrix}]

Does not exist

[\begin{matrix} 1 & - 1 \\ 0 & 1 \end{matrix}]

[\begin{matrix} - 1 & - 1 \\ 0 & - 1 \end{matrix}]

Verify by multiplication:

$A A^{- 1} = I$

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A matrix is invertible if and only if:

Its determinant is non-zero

It is square

It is symmetric

It is diagonal

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The inverse of

$[\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}]$

is:

[\begin{matrix} 0.5 & 0 \\ 0 & \frac{1}{3} \end{matrix}]

[\begin{matrix} - 2 & 0 \\ 0 & - 3 \end{matrix}]

[\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}]

Does not exist

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$A$

and

$B$

are invertible, then

$(A B)^{- 1} =$

A^{- 1} B^{- 1}

A B

B A

B^{- 1} A^{- 1}

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The inverse of a rotation matrix

$[\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]$

is:

[\begin{matrix} - \cos θ & \sin θ \\ - \sin θ & - \cos θ \end{matrix}]

[\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix}]

[\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]

Does not exist

Rotation matrices are orthogonal, so

$A^{- 1} = A^{T}$

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$A^{2} = I$

, then

$A^{- 1} =$

A^{2}

I

A

0

$A \cdot A = I$

implies

$A^{- 1} = A$

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The inverse of a symmetric matrix is:

Symmetric

Skew-symmetric

Diagonal

Zero

$(A^{- 1})^{T} = (A^{T})^{- 1} = A^{- 1}$

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The matrix $[\begin{matrix} 1 & 2 \\ 2 & 4 \end{matrix}]$ is:

Not invertible

Invertible

Orthogonal

Nilpotent

Its determinant is $(1) (4) - (2) (2) = 0$

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$A$

is invertible, then

$(A^{T})^{- 1} =$

A

A^{- 1}

(A^{- 1})^{T}

A^{T}

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The inverse of an elementary matrix is:

Another elementary matrix

A zero matrix

A diagonal matrix

Not guaranteed to exist

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The cofactor of element 2 in $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$ is:

-4

-3

Cofactor $= (- 1)^{1 + 2} \cdot 3 = - 3$

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The adjoint of

$[\begin{matrix} a & b \\ c & d \end{matrix}]$

is:

[\begin{matrix} a & c \\ b & d \end{matrix}]

[\begin{matrix} d & - b \\ - c & a \end{matrix}]

[\begin{matrix} d & b \\ c & a \end{matrix}]

[\begin{matrix} - a & - b \\ - c & - d \end{matrix}]

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For a

$3 \times 3$

matrix, the cofactor of

$a_{i j}$

is:

(- 1)^{i + j}

times the minor

The minor itself

(- 1)^{i j}

times the minor

Zero

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The adjoint of a diagonal matrix is:

Zero

Identity

Diagonal

Undefined

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$A$

$2 \times 2$

, then

$A \cdot adj (A) =$

I

\det (A) \cdot I

A^{T}

Zero

For any square matrix,

$A \cdot adj (A) = \det (A) \cdot I$

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The minor of an element is the determinant of:

Matrix after deleting its row and column

The transposed matrix

The squared matrix

Matrix with added row

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The cofactor matrix of

$I$

₂ is:

I

₂

[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]

[\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}]

[\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]

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$A$

is invertible, then

$adj (A^{- 1}) =$

A

adj (A)

\det (A) \cdot A

\frac{adj (A)}{\det (A)}

$adj (A^{- 1}) = \frac{adj (A)}{\det (A)}$

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The adjoint of a singular matrix is:

Identity

Invertible

Zero

Not defined

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The cofactor expansion is used to compute:

Matrix addition

Determinants

Eigenvalues

Matrix multiplication

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The system {2x+3y=5, 4x+6y=10} has:

A unique solution

No solution

Infinitely many solutions

Exactly two solutions

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Cramer’s Rule applies to systems where:

Number of equations exceeds variables

Coefficient matrix is singular

System is homogeneous

Coefficient matrix is square and invertible

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The system {x+y=2, 2x+2y=5} has:

A unique solution

No solution

Infinitely many solutions

Exactly two solutions

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If A is the coefficient matrix of AX=B, the system has a unique solution if:

det(A)=0

A is singular

B=0

det(A)≠0

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The solution to {x-y=1, 2x+y=5} is:

(2,1)

(1,2)

(3,1)

(1,3)

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A homogeneous system AX=0 always has:

No solution

A unique solution

Infinitely many solutions

At least one solution (X=0)

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The inverse method for solving AX=B gives:

X=A⁻¹B

X=B⁻¹A

X=AB⁻¹

X=BA⁻¹

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The system {x+2y=3, 3x+6y=9} is:

Consistent and independent

Consistent and dependent

Inconsistent

Nonlinear

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For a system AX=B, if rank(A)=rank([A|B]) < n, the system has:

A unique solution

No solution

Infinitely many solutions

Exactly two solutions

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The number of solutions to {x+y+z=1, 2x+2y+2z=2} is:

Infinite

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Matrices are used in computer graphics to perform:

Data sorting

Geometric transformations

File compression

Encryption

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In Markov chains, the transition matrix is:

Symmetric

Stochastic (rows sum to 1)

Diagonal

Skew-symmetric

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The adjacency matrix of a graph is always:

Symmetric

Diagonal

Singular

Upper triangular

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In solving linear differential equations, matrices are used to:

Plot graphs

Compute integrals

Represent systems of equations

Differentiate functions

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The Leslie matrix is used in:

Population modeling

Image processing

Cryptography

Quantum mechanics

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The Jacobian matrix is applied in:

Solving PDEs

Change of variables in integration

Graph theory

Probability

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In economics, input-output models use matrices to:

Predict stock prices

Model supply-demand

Calculate GDP

Analyze production sectors

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The Hessian matrix is used in:

Optimization (identifying extrema)

Network analysis

Machine learning

Fluid dynamics

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In quantum mechanics, matrices represent:

Wave functions

Observables (e.g., spin)

Probabilities

Energy levels

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The rotation matrix in 2D for angle θ is:

[\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]

[\begin{matrix} \sin θ & \cos θ \\ \cos θ & - \sin θ \end{matrix}]

[\begin{matrix} 1 & \tan θ \\ - \tan θ & 1 \end{matrix}]

[\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix}]

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