Matrices: For what value of c is it true that AB = I ?

[Angela123]

A=[1 -4 B=[-3 4
1 -3] -1 c]

For what value of c is it true that AB=I? c =?

I multiplied the two together and got: [1 (4-4c)
0 (4-3c)]
4-4c=1 c=3/4
4-3c=0 c=4/3
Did I do the last step wrong? There is only supposed to be one answer.

 

[mmm4444bot]

Re: Matrices

A=[1 -4 B=[-3 4
1 -3] -1 c]

For what value of c is it true that AB=I? c =?

I multiplied the two together and got: [1 (4-4c)
0 (4-3c)]
4-4c=1 c=3/4
4-3c=0 c=4/3
Did I do the last step wrong? There is only supposed to be one answer.

For the Identity Matrix, you mixed up the positions of the values 1 and 0 highlighted in red above. Swap them, and resolve those two equations for c.

MY EDITS: PS: If you want to "draw" matrices, then please use the [ code ] and [ /code ] tags as shown below. (You can click the Quote button to see this coding as I typed it.)

A = 1 -4    B = -3 4
    1 -3        -1 c

 

[Deleted member 4993]

Re: Matrices

    A = [ 1 -4 ]   B = [ -3 4 ]
        [ 1 -3 ]       [ -1 c ]

For what value of c is it true that AB=I? c =?

I multiplied the two together and got: [1 (4-4c)
0 (4-3c)]
4-4c=1 c=3/4 <<<<<<<This is wrong

4 - 4c = 0
4c = 4
c = 1

4-3c=0 c=4/3<<<<<<<This is wrong

 

[Angela123]

So if you are looking for the identity matrix, then you make the equation equal to the number diagonal from it?

 

[Deleted member 4993]

So if you are looking for the identity matrix, then you make the equation equal to the number diagonal from it?

No--

Identity matrix has 1 as the diagonal number - and rest are zeros. Incidentally diagonal of a [square] matrix - by definition runs from top left to bottom right.

 

[mmm4444bot]

So if you are looking for the identity matrix, then you make the equation equal to the number diagonal from it?

I do not understand to what you refer by using the words highlighted in red, so I am unable to understand your question.

You posted that you got the following result from multiplying matrix A times matrix B.

$\displaystyle \left[ \begin{array}{cc} 1 & -4 & 1 & -3 \end{array} \right] \cdot \left[ \begin{array}{cc} -3 & 4 & -1 & c \end{array} \right] \;=\; \left[ \begin{array}{cc} 1 & 4-4c & 0 & 4 - 3c \end{array} \right]$

You want the value of c that makes this product a two-by-two Identity Matrix.

$\displaystyle \left[ \begin{array}{cc} 1 & -4 & 1 & -3 \end{array} \right] \cdot \left[ \begin{array}{cc} -3 & 4 & -1 & c \end{array} \right] \;=\; \left[ \begin{array}{cc} 1 & 0 & 0 & 1 \end{array} \right]$

I tried to tell you in my response that you set the expression 4 - 4c equal to 1, instead of zero, and that you set the expression 4 - 3c equal to zero, instead of 1.

If you fix these two mistakes in your equations, then your approach will work.

Do you know the definition for "Identity Matrix"?

~ Mark



MY EDIT: corrected erroneous dimension 'four-by-four'

 

[Angela123]

I was under the impression that the identity matrix is the inverse of A. What I meant by my question was, after I have my equation(s), how do I know what to set them equal to in order to find c? I can see how you got the answer for this one, but suppose it was a 3x3 or 3x2?

 

[Deleted member 4993]

I was under the impression that the identity matrix is the inverse of A. What I meant by my question was, after I have my equation(s), how do I know what to set them equal to in order to find c? I can see how you got the answer for this one, but suppose it was a 3x3 or 3x2?

Identity matrix is always a square matrix - so [3 x 2] is not possible.

[3 x 3] matrix will be similar - it will have three 1's down the diagonal and 6 zeros every where else.

 

[mmm4444bot]

I was under the impression that the identity matrix is the inverse of A.

Nope. The inverse of matrix A is called the Inverse Matrix of A.

The Identity Matrix contains only 1s and 0s.

Subhotosh gave you its definition.

Identity matrix has 1 as the diagonal number - and rest are zeros. Incidentally diagonal of a [square] matrix - by definition runs from top left to bottom right.

Look again at the equations that I posted earlier. Can you see by comparing the corresponding positions between the Identity Matrix and the one above it (that contains your expressions with c) that the expression 4 - 4c must equal zero?

In other words, the expression 4 - 4c is in Row 1, Column 1.

In the Identity Matrix, there is a zero in Row 1, Column 1.

Cheers,

~ Mark