Size of AB

Let be a matrix $val18, a matrix $val19.

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 $val24 matrix. Is the following statement true?

If $val21, $val22.


Invertibility

$val25. Is the following statement true?

If $val20, then $val21.

( is the $val18×$val18 identity matrix.)


Matrix operations

Consider two matrices

.

Does make sense?
Does make sense?
Does make sense?
Does make sense?
Does make sense?


Multiplicability

$val19$val17. Is the following statement true?

If $val14, then $val15.


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 $val6×$val6 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 $val6×$val6 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 $val6×$val6 matrices, and two numbers. Among the following properties, which ones are $val23?

1.$val22
2.$val22
3.$val22
4.$val22


Algebraic properties II

$val28. Is the following property true?

$val26

( is the $val22×$val22 identity matrix.)


Properties of multiplication

$val21 $val15×$val15. Is the following statement true?

If $val17, then $val18.

( is the $val15×$val15 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.