For Addition :Matrix addition requires that all matrices involved be of the same size. Specifically, matrices must have identical dimensions (e.g., 3 × 3, 1 × 2, meaning matrix A is 2×2 then matrix B must be 2×2 ). Adding matrices of different sizes, such as 2 × 3 and 3 × 2, is not possible.
For Subtraction:Matrix subtraction also requires that all matrices involved have the same dimensions. Only matrices with identical sizes, such as 4 × 4 or 2 × 5, meaning matrix A is 2×2 then matrix B must be 2×2 can be subtracted from one another. Subtracting matrices of different sizes, like a 3 × 2 and a 2 × 3, is not allowed.
For Multiplication To multiply two matrices, the number of columns in the first matrix must be the same as the number of rows in the second matrix. For example, you can multiply a 2 × 3 matrix by a 3 × 4 matrix, but you can't multiply a 2 × 3 matrix by a 4 × 3 matrix.
For Transpose The transpose of a matrix is an operation where you flip the matrix over its diagonal. This means that the rows become columns and the columns become rows. For example, if you have an element at position \(a_{ij}\) in the original matrix, it will move to position \(a_{ji}\) in the transposed matrix. This is usually indicated by adding a "T" as an exponent, like \(A^T\). So, the first row of the original matrix becomes the first column of the transposed matrix, and so on.