Explain the notation M^-1 used for the inverse of a square matrix M.

I thought M^-1 would be a normal inverse matrix except the numbers would be vertically switched like in fractions...? Not sure though.

If M is 2x2 with all ones, and N is 2x2 with all one-half, M times N will not give the identity matrix.

The notation M^-1 represents the inverse of a square matrix M. The inverse of a matrix is a matrix that, when multiplied by the original matrix, yields the identity matrix. The identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere.

To find the inverse of a matrix M, you need to follow certain steps:

1. Ensure that M is a square matrix. The inverse can only be found for square matrices, which have the same number of rows and columns.

2. Calculate the determinant of M. The determinant is a scalar value that can help determine if the matrix is invertible. If the determinant is zero, then the matrix does not have an inverse.

3. If the determinant is non-zero, continue with the next steps. Otherwise, the matrix does not have an inverse.

4. Find the adjugate of M. The adjugate matrix is obtained by transposing the matrix of cofactors. The cofactor of an element in a matrix is calculated by taking the determinant of the submatrix formed by removing the row and column of that element and multiplying it by (-1)^(i+j), where i and j are the row and column indices, respectively.

5. Divide the adjugate matrix by the determinant of M. This step involves dividing each element of the adjugate matrix by the determinant value.

The resulting matrix is the inverse of M, denoted as M^-1. Multiplying a matrix by its inverse should yield the identity matrix, denoted as I.

So, the notation M^-1 does not indicate switching numbers vertically like in fractions. It represents the inverse of a square matrix calculated using the steps mentioned above.

The notation M^-1 represents the inverse of a square matrix M. The inverse of a matrix has the property that when multiplied with the original matrix, it yields the identity matrix (I). In other words, if M^-1 is the inverse of M, then M^-1 * M = I, where I is the identity matrix.

To calculate the inverse of a square matrix M, there are various methods, such as the Gauss-Jordan elimination method or the matrix adjugate method.

One common way to represent a matrix is by using rows and columns. So, if we have a matrix M:

M = [a b]
[c d]

The inverse matrix M^-1 would then be:

M^-1 = [x y]
[z w]

The values of x, y, z, and w are calculated such that M^-1 * M = I.

The formula to calculate the inverse matrix is:

M^-1 = (1 / determinant of M) * adjugate of M,

where the adjugate of M is obtained by swapping the elements of the main diagonal (a and d) and changing the sign of the other two elements (b and c).

So, in summary, the notation M^-1 represents the inverse of a square matrix M, which is a matrix that, when multiplied with the original matrix, gives the identity matrix. The actual values of the inverse matrix are obtained by using formulas and calculations based on the matrix elements.

It involves much more than that.

I suggest you try google, where you can find full explanations.