How to prove or disprove
(a)if A has a zeronentryonthe diagonal then A is not invertible
(b)if Ais not invertible then for every matrix B, AB is not invertible
(c)if A is a nonzero 2X2 matrix such that A^2+A=0, then A is invertible
To prove or disprove these statements, we need to use logical reasoning and knowledge of matrix properties. Let's break down each statement and explain how to prove or disprove it:
(a) Statement: If matrix A has a zero entry on the diagonal, then A is not invertible.
To prove this statement, we need to show that if A has a zero entry on the diagonal, it will not have an inverse.
Proof:
1. Assume A is invertible and has a zero entry on the diagonal.
2. By the definition of an inverse, if A is invertible, there exists a matrix B such that AB = BA = I (the identity matrix).
3. Multiplying matrix A by matrix B, we get AB = 0 (since there is a zero entry on the diagonal).
4. However, this contradicts the definition of the identity matrix, where AB = I. Thus, our assumption that A is invertible is incorrect.
5. Therefore, if matrix A has a zero entry on the diagonal, it is not invertible.
(b) Statement: If matrix A is not invertible, then for every matrix B, AB is not invertible.
To prove this statement, we need to show that if A is not invertible, any product AB (where B is another arbitrary matrix) is also not invertible.
Proof:
1. Let A be a non-invertible matrix, which means there does not exist a matrix B such that AB = BA = I.
2. Assume there exists a matrix B such that AB is invertible.
3. If AB is invertible, there exists a matrix C such that (AB)C = I.
4. Multiplying both sides by A, we have A(BC) = I.
5. This implies that A is invertible, which contradicts our initial assumption.
6. Therefore, if matrix A is not invertible, then for every matrix B, AB is not invertible.
(c) Statement: If matrix A is a nonzero 2x2 matrix such that A^2 + A = 0, then A is invertible.
To prove this statement, we need to show that if A is a nonzero 2x2 matrix satisfying A^2+A=0, then A is invertible.
Proof:
1. Assume A is a nonzero 2x2 matrix such that A^2 + A = 0.
2. If A is invertible, then there exists a matrix B such that AB = BA = I.
3. Multiplying both sides of the given condition by A, we have A^3 + A^2 = 0.
4. Rearranging the equation, we get A^3 = -A^2.
5. By left-multiplying both sides of the equation by A^(-1) (inverse of A), we have A^2 = -I. Here, I denotes the identity matrix.
6. However, the condition given states that A^2 + A = 0. Therefore, -I + A = 0.
7. Adding A to both sides, we get A - I + A = 0.
8. Rearranging, we have 2A - I = 0.
9. Multiplying by A^(-1), we have A = (1/2)I, which implies that A is a scalar matrix.
10. A scalar matrix is invertible if and only if the scalar is nonzero.
11. Since A is nonzero (by the condition given), it follows that A is invertible.
12. Therefore, if matrix A is a nonzero 2x2 matrix such that A^2 + A = 0, then A is invertible.
By following the logical steps outlined in the proofs, we can either prove or disprove these statements based on the assumptions and conditions given.