Questions: Let A=[1 2; 3 -1]. Compute the Frobenius norm AF 5 15 sqrt(15) sqrt(5)

Transcript text: Let $A=\left[\begin{array}{cc}1 & 2 \\ 3 & -1\end{array}\right]$. Compute the Frobenius norm $\|A\|_{F}$ 5 15 $\sqrt{15}$ $\sqrt{5}$ SUBMIT

Solution

Solution Steps

To compute the Frobenius norm of a matrix, we need to take the square root of the sum of the absolute squares of its elements. For the given matrix $ A $, calculate the sum of the squares of each element, and then take the square root of that sum.

Step 1: Define the Frobenius Norm

The Frobenius norm of a matrix $ A $ is defined as: \[ \|A\|_F = \sqrt{\sum_{i=1}^{m} \sum_{j=1}^{n} |a_{ij}|^2} \] where $ a_{ij} $ are the elements of the matrix $ A $.

Step 2: Calculate the Sum of Squares of Elements

For the matrix $ A = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix} $, calculate the sum of the squares of its elements: \[ 1^2 + 2^2 + 3^2 + (-1)^2 = 1 + 4 + 9 + 1 = 15 \]

Step 3: Compute the Frobenius Norm

Take the square root of the sum obtained in Step 2: \[ \|A\|_F = \sqrt{15} \approx 3.872983346207417 \]