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

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.

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 \( A \).

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

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

Final Answer