Understanding Eigenvalues | The Truth about Real Eigenvalues for 2 × 2 Matrices

Every 2 × 2 matrix has at least one real eigenvalue. (T/F)

The statement “Every 2 × 2 matrix has at least one real eigenvalue” is true

The statement “Every 2 × 2 matrix has at least one real eigenvalue” is true.

To understand this, we need to first define what eigenvalues are. In linear algebra, eigenvalues are scalar values that represent the factors by which a matrix “stretches” or “compresses” a vector when multiplied by it. In other words, given a square matrix A and a vector v, when A is multiplied by v, the resulting vector may be scaled by some factor λ, which is the eigenvalue associated with that vector.

Now, for a 2 × 2 matrix, we can define it as:

A = [[a, b],
[c, d]],

where a, b, c, and d are real numbers. To find the eigenvalues of this matrix, we need to solve the characteristic equation:

det(A – λI) = 0,

where λ is the eigenvalue we are trying to find, and I is the 2 × 2 identity matrix.

For a 2 × 2 matrix, the characteristic equation becomes:

det([[a – λ, b],
[c, d – λ]]) = 0.

Expanding this determinant, we have:

(a – λ)(d – λ) – bc = 0,
ad – λ(a + d) + λ^2 – bc = 0,
λ^2 – λ(a + d) + (ad – bc) = 0.

This is a quadratic equation in λ. We can solve it using the quadratic formula:

λ = (-b ± sqrt(b^2 – 4ac)) / 2a.

Applying this to the equation, we obtain:

λ = [(a + d) ± sqrt((a + d)^2 – 4(ad – bc))] / 2.

To determine whether the eigenvalues we obtain are real or complex, we need to consider the discriminant inside the square root:

D = (a + d)^2 – 4(ad – bc).

If D is positive, then the eigenvalues are real. If D is negative, then the eigenvalues are complex. If D is zero, then the eigenvalues are repeated (real, equal eigenvalues).

Now, plugging in the values from the matrix A:

D = (a + d)^2 – 4(ad – bc) = a^2 + 2ad + d^2 – 4ad + 4bc = (a – d)^2 + 4bc.

Since we know that a, b, c, and d are real numbers, the square of any real number is nonnegative. Therefore, (a – d)^2 is nonnegative. Similarly, 4bc is also nonnegative. So, the sum of these nonnegative quantities is always nonnegative, implying that D is nonnegative as well.

This means that the discriminant D is either zero or positive, indicating that the eigenvalues obtained from the quadratic equation are either real or repeated real eigenvalues. Therefore, every 2 × 2 matrix has at least one real eigenvalue.

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