what is invertible matrices
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In linear algebra, an n-by-n square matrix A is called invertible if there exists an n-by-n square matrix B such that where Iₙ denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.
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An invertible matrix is a matrix M such as there exists a matrix N such as MN=NM=In.
Looking at this equation, it is clear that this equation can only stand if M is an n×n square matrix. N is therefore noted M−1.
So, what exactly does that mean?
We can consider a matrix M∈Mn(R) as a linear application μ:Rn→Rn. Using the canonical base, the image by μ of x=x1e1+…+xnen is
μ(x)=∑i=1n∑j=1nMi,jxjei
You can consider the inverse of the matrix M as the inverse of the function μ:
M−1MX=InX=X
μ−1(μ(x))=x
The inverse is especially useful to solve linear equation: if you have the following general linear equation MX=Y, if you want to solve this equation, you just have to invert the matrix M, and then you have M−1MX=X=M−1Y.
Please notice that a matrix is not always invertible: for example, the matrix 0n filled with zeros is not invertible. More generally, a matrix is invertible if and only if its determinant is not equal to 0, which is equivalent to say that its rank is n.
Looking at this equation, it is clear that this equation can only stand if M is an n×n square matrix. N is therefore noted M−1.
So, what exactly does that mean?
We can consider a matrix M∈Mn(R) as a linear application μ:Rn→Rn. Using the canonical base, the image by μ of x=x1e1+…+xnen is
μ(x)=∑i=1n∑j=1nMi,jxjei
You can consider the inverse of the matrix M as the inverse of the function μ:
M−1MX=InX=X
μ−1(μ(x))=x
The inverse is especially useful to solve linear equation: if you have the following general linear equation MX=Y, if you want to solve this equation, you just have to invert the matrix M, and then you have M−1MX=X=M−1Y.
Please notice that a matrix is not always invertible: for example, the matrix 0n filled with zeros is not invertible. More generally, a matrix is invertible if and only if its determinant is not equal to 0, which is equivalent to say that its rank is n.
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