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Q: Do all matrices have determinant

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A determinant is defined only for square matrices, so a 2x3 matrix does not have a determinant.Determinants are defined only for square matrices, so a 2x3 matrix does not have a determinant.

The determinant is only defined for square matrices.

I assume since you're asking if 2x2 invertible matrices are a "subspace" that you are considering the set of all 2x2 matrices as a vector space (which it certainly is). In order for the set of 2x2 invertible matrices to be a subspace of the set of all 2x2 matrices, it must be closed under addition and scalar multiplication. A 2x2 matrix is invertible if and only if its determinant is nonzero. When multiplied by a scalar (let's call it c), the determinant of a 2x2 matrix will be multiplied by c^2 since the determinant is linear in each row (two rows -> two factors of c). If the determinant was nonzero to begin with c^2 times the determinant will be nonzero, so an invertible matrix multiplied by a scalar will remain invertible. Therefore the set of all 2x2 invertible matrices is closed under scalar multiplication. However, this set is not closed under addition. Consider the matrices {[1 0], [0 1]} and {[-1 0], [0 -1]}. Both are invertible (in this case, they are both their own inverses). However, their sum is {[0 0], [0 0]}, which is not invertible because its determinant is 0. In conclusion, the set of invertible 2x2 matrices is not a subspace of the set of all 2x2 matrices because it is not closed under addition.

The square matrix have determinant because they have equal numbers of rows and columns. <<>> Determinants are not defined for non-square matrices because there are no applications of non-square matrices that require determinants to be used.

For small matrices the simplest way is to show that its determinant is not zero.

actually MATRICES is the plural of matrix which means the array of numbers in groups and columns in a rectangular table... and determinant is used to calculate the magnitude of a matrix....

It isn't clear what you want to solve for. If you want to find the matrix, there is not a unique solution - there are infinitely many matrices with the same determinant.

No. Determinants are only defined for square matrices.No. Determinants are only defined for square matrices.

That's a special calculation done on square matrices - for example, on a 2 x 2 matrix, or on a 3 x 3 matrix. For details, see the Wikipedia article on "Determinant".

If you mean what does something like SL(3, R) mean, it is the group of all 3X3 matrices with determinant 1, with real entries, under matrix multiplication.

If R is a commutative ring with unit, then the commutator subgroup of GLn(R) is SLn(R), the special linear group, which consists of all matrices in GLn(R) with determinant 1.

relationship between determinant and adjoint

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