What does it mean when the determinant of a matrix is 0?

What does it mean when the determinant of a matrix is 0?

When the determinant of a matrix is zero, the volume of the region with sides given by its columns or rows is zero, which means the matrix considered as a transformation takes the basis vectors into vectors that are linearly dependent and define 0 volume.

How do you know if a determinant will be zero?

If either two rows or two columns are identical, the determinant equals zero. If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.

What does a non zero determinant mean?

If a matrix has a non-zero determinant, then it has an inverse, and that inverse multiplied by a vector will always give an answer, so in order for a system of equations to have no answer, the matrix cannot have an inverse, so the determinant must be zero.

When using Cramer’s rule if D 0 then the system of linear equations is inconsistent?

When the determinant of the coefficient matrix D is zero, the formulas of Cramer’s rule are undefined. In this case, the system is either dependent or inconsistent depending on the values of Dx and Dy. When D=0 and both Dx=0 and Dy=0 the system is dependent.

Does determinant 0 mean linearly dependent?

Since the matrix is , we can simply take the determinant. If the determinant is not equal to zero, it’s linearly independent. Otherwise it’s linearly dependent. Since the determinant is zero, the matrix is linearly dependent.

Does the equation AB I imply that A is invertible?

Theorem. Let A be a square matrix. If B is a square matrix such that either AB = I or BA = I, then A is invertible and B = A−1.

Is det AB )= det A det B?

The proof is to compute the determinant of every elementary row operation matrix, E, and then use the previous theorem. det(AB) = det(A) det(B). Proof: If A is not invertible, then AB is not invertible, then the theorem holds, because 0 = det(AB) = det(A) det(B)=0.

What is the determinant of 3A?

3A is the matrix obtained by multiplying each entry of A by 3. Thus, if A has row vectors a1, a2, and a3, 3A has row vectors 3a1, 3a2, and 3a3. Since multiplying a single row of a matrix A by a scalar r has the effect of multiplying the determinant of A by r, we obtain: det(3A)=3 · 3 · 3 det(A) = 27 · 2 = 54.

How do you find the determinant of a 3 by 3 matrix?

To work out the determinant of a 3×3 matrix:

1. Multiply a by the determinant of the 2×2 matrix that is not in a’s row or column.
2. Likewise for b, and for c.
3. Sum them up, but remember the minus in front of the b.