The Matrix A is
(i) Find the matrix such that where is the identity matrix.
(ii) Find the inverse matrix
(iii) Use to solve the simultaneous equations
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The Identity Matrix is a square matrix with 1's down the main diagonal and 0's everywhere else.
It is usually abbreviated to , and you have to remember what dimensions your matrices are. In your case, your matrices are , so you will be using .
It is the matrix equivalent of the number , because when you premultiply or postmultiply any matrix by , you get that original matrix.
Also, if you multiply any nonsingular matrix by its inverse, you get the identity matrix. This gives us the ability to perform the matrix equivalent of division.
First, notice that you can rewrite the system of equations as
Write the system in matrix form.
I am going to write this equation as
.
Premultiply both sides of the equation by .
Then you have
.
Recall that if you multiply any matrix by its inverse, you get the identity matrix, so
And if you multiply any matrix by the identity matrix, you get back to original matrix.
So
Therefore and .
How can you be given questions on these things without having been taught them? Matrix addition, Matrix multiplication and the Identity Matrix are the most important things to know when dealing with matrices.
Matrix addition is performed component-wise. So you add (or subtract) corresponding components.
So in your case
No, an identity matrix is not equal to 1. It ACTS LIKE a 1 when you are dealing with matrices.
And you can't ALWAYS use because the matrices you are working with might not be . It depends on the dimensions of the matrices you are working with. In this case you use because the matrices you are working with are .
But what if you were dealing with matrices?
Which would you use then?