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Quizz matrices

Quizz matrices
--- Introduction ---

This module contains 17 training exercises on basic notions of
matrices: multiplication, inverse, trace, rank. See
Quizz determinant for similar exercises on the determinant.
Students may use them as a training tool for a better memorization of the lesson,
while teachers may put them into a work sheet with activated time limit, so that
proficiency can be tested.

Size of AB

Let
be a matrix ,
a matrix .

Give the condition for
to make sense.

In this case,
has
rows and
columns.

ABA

We have a matrix
, such that the multiplication

makes sense.

What is the size of
?
×

What is the size of
?
×

ABC

We have a matrix
, such that the multiplication

makes sense.

What is the size of
?
rows and
columns.

Invertibility and coefficients

Let
be a matrix. Is the following statement true?

If , .

Invertibility

. Is the following statement true?

If , then .

(
is the × identity matrix.)

Matrix operations

Consider two matrices

.

Does
make sense ?

Does
make sense ?

Does
make sense ?

Does
make sense ?

Does
make sense ?

Multiplicability

. Is the following statement true?

If , then .

Mult-one 4x4

What is the value of
in the following multiplication of matrices?

Partial multiplication 3x3

We have an equation of multiplication of matrices × as follows, where the question marks represent unknown coefficients.

There is only one determinable coefficient in the product matrix. It is
.

Type c11 for
for example.

Partial multiplication 4x4

We have an equation of multiplication of matrices × as follows, where the question marks represent unknown coefficients.

There is only one determinable coefficient in the product matrix. It is
.

Type c11 for
for example.

Algebraic properties

Let
be three × matrices, and
and
two numbers. Among the following properties, which ones are ?

Algebraic properties II

. Is the following property true?

(
is the × identity matrix.)

Properties of multiplication

×. Is the following statement true?

If , then .

(
is the × identity matrix.)

Trace 2x2

What is the trace of the matrix

?

Trace 3x3

What is the trace of the matrix

?

Trace 4x4

What is the trace of the matrix

?

Triangular invertibility 3x3

Determine whether the following matrix is invertible, where
is a real number.

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Description: elementary questions on matrices. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games