Matrices

[[1, 2], [2, 3]], [[5, 0], [0, -2]], [[1, 3, 2], [3, 0, 4], [2, 4, 1]]

A matrix is symmetric if A = A^T (transpose). Check each: Option 1: A^T = [[1, 2], [2, 3]] = A ✓. Option 2: A^T = [[0, -1], [1, 0]] ≠ A ✗. Option 3: A^T = [[5, 0], [0, -2]] = A ✓. Option 4: A^T = [[1, 3, 2], [3, 0, 4], [2, 4, 1]] = A ✓. Option 5: A^T = [[2, 3], [1, 2]] ≠ A ✗.

3 × 4

The order of a matrix is given as rows × columns. Matrix A has 3 rows and 4 columns, so its order is 3 × 4. Common mistake: Confusing rows with columns or writing columns × rows instead of rows × columns.

[[3, 0], [0, 5]], [[7, 0, 0], [0, -2, 0], [0, 0, 1]], [[0, 0], [0, 0]]

A diagonal matrix has all non-diagonal elements as zero. Option 1: All off-diagonal elements are 0 ✓. Option 2: Has 2 in position (1,2) ✗. Option 3: All off-diagonal elements are 0 ✓. Option 4: Zero matrix is also diagonal ✓. Option 5: Has 1 in position (1,2) ✗.

4

The transpose of A = [[1, 2], [3, x]] is A^T = [[1, 3], [2, x]]. Given that A^T = [[1, 3], [2, 4]], we can compare: The element in position (2,2) of A^T is x, which equals 4. Therefore, x = 4.