Contents
Does the determinant distribute?
determinant: The unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of 1 for the unit matrix. Its abbreviation is “det “.
What is detA detB?
Theorem 1: If A and B are both n × n matrices, then detAdetB = det(AB). Theorem 2: A square matrix is invertible if and only if its determinant is non-zero. A. The proof of Theorem 2. Use the multiplicative property of determinants (Theorem 1) to give a one line proof that if A is invertible, then detA = 0.
Does det A det (- A?
If A is an n×n square matrix and n is odd, then det(−A)=−det(A).
How do you know if a determinant is correct?
Here are the steps to go through to find the determinant.
- Pick any row or column in the matrix. It does not matter which row or which column you use, the answer will be the same for any row.
- Multiply every element in that row or column by its cofactor and add. The result is the determinant.
Why is AB not invertible?
If B is not invertible, it has a non-trivial kernel. Take a vector from it and apply AB. I see. So then AB has a non-trivial kernel, which means that AB is not invertible.
How do you solve Det AB?
If A and B are n × n matrices, then det(AB) = (detA)(detB). In other words, the determinant of a product of two matrices is just the product of the deter- minants. from the previous example.
Does det A det A 2?
No. Hence determinant of A^2 is equal to square of Determinant of A !
How is det calculated?
The determinant of a matrix is a special number that can be calculated from a square matrix….To work out the determinant of a 3×3 matrix:
- Multiply a by the determinant of the 2×2 matrix that is not in a’s row or column.
- Likewise for b, and for c.
- Sum them up, but remember the minus in front of the b.
When does det AB ) = Deta detb?
If A and B are n × n matrices, then det(AB) = (detA)(detB). In other words, the determinant of a product of two matrices is just the product of the deter- minants.
Which is the formula for det ( A + B )?
For even n, let A = − B and det (A) > 0, so det (A+B) = 0< det (A)+ det (B). Now, consider A = B. We have det (A+ B) = det (2A) = 2n det (A) > 2 det (A) = det (A)+ det (B) for n> 1 and det (A) > 0. Thus, either inequality can hold. In general, you can’t expect a formula for det (A+B).
How to prove that det AB is not invertible?
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. Suppose that A is invertible.
When does det ( A + B ) hold in linear algebra?
Then det ( A + B) = det ( 2 I n) = 2 n det ( I n) = 2 n and det ( A) + det ( B) = 1 + 1 = 2 so for n > 1, your equality does not hold at least for these matrix. And for n = 1, since the determinant is the only element of the matrix, we do have your equality.