Is the Matrix A with Entries aij=min(i,j) Always Invertible?

[RobandJoanna]

Help! System of equations!

For a natural number define the n*n matrix A = (aij) by setting aij = min(i,j). Write down the matrix a for a few values of n = 2,3,4... Show that A is invertible and find its inverse.
Hence or otherwise show that the system of equations Ax =b , where x=(x1,...xn) and b=(b1...bn) has a unique solution.

Anyone help with how to solve that?
Thanks
Rob and Joanna!

 

[shmoe]

Have you tried n=2, 3, and 4 at least?

Write them down and try to reduce to the identity by row operations. Should be easy. You shold be able to find inverse in each case too.

 

[M98Ranger]

Hi Rob and Joanna,

Solving systems of equations can be challenging, but with some practice and understanding of certain concepts, it can become easier. Let's first start by defining what a system of equations is. A system of equations is a set of two or more equations with multiple variables that need to be solved simultaneously.

In this case, we have a system of equations represented by the matrix A and vector x. The first step in solving this system is to understand the properties of the given matrix A. From the given definition, we can see that A is a lower triangular matrix, which means that all the entries above the main diagonal are zero.

For n = 2, the matrix A would look like this:

A = [1 0
0 2]

Similarly, for n = 3, the matrix A would be:

A = [1 0 0
0 2 0
0 0 3]

We can see that for any value of n, the matrix A will always be invertible. This is because the determinant of a lower triangular matrix is equal to the product of its diagonal entries, which in this case is always greater than zero. Therefore, we can find the inverse of A using the formula:

A^-1 = 1/det(A) * adj(A)

Where adj(A) is the adjugate matrix of A. Using this formula, we can find the inverse of A for any value of n.

Now, coming to the system of equations Ax = b, we can see that the matrix A is invertible, which means that it has a unique solution. This is because the inverse of A exists and is unique. We can find the solution by multiplying both sides of the equation by A^-1, giving us:

x = A^-1 * b

Using the inverse we calculated earlier, we can easily find the solution to this system of equations.

I hope this helps in understanding how to solve systems of equations using matrices. If you have any further doubts, please let me know. Keep practicing and you'll become a pro in no time!