This demonstration is detailed below using a generic diagonal matrix:Ī diagonal matrix is invertible if, and only if, all elements on the main diagonal are different from 0.Īlso, the inverse of a diagonal matrix will always be another diagonal matrix with the reciprocals of the numbers on the main diagonal:įrom the previous characteristic, it can be deduced that the determinant of the inverse of a diagonal matrix is equal to the product of the reciprocals of the entries on the main diagonal:Īs we have seen, solving calculations with diagonal matrices is very simple, since many zeros are involved in the operations. This theorem is easy to prove: we only have to calculate the determinant of a diagonal matrix by cofactors. Look at the following solved exercise in which we find the determinant of a diagonal matrix by multiplying the elements on its main diagonal: The determinant of a diagonal matrix is the product of the elements on the main diagonal. To calculate the power of a diagonal matrix we must raise each element of the diagonal to the exponent: To solve a multiplication or a matrix product of two diagonal matrices we just have to multiply the elements of the diagonals with each other. The addition (and subtraction) of two diagonal matrices is very simple: you just have to add (or subtract) the numbers of the diagonals. Addition and subtraction of diagonal matrices That is why they are so used in mathematics. One of the reasons that diagonal matrices are so important to linear algebra is the ease with which they allow you to perform calculations. See: formula for adjoint of a matrix Operations with diagonal matrices
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