# 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 two numbers. Among the following properties, which ones are ?

 1 2 3 4

### 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.

Other exercises on: matrices   determinant   linear algebra