Question

Find \(A B\), if possible. $$ A=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & -2 \end{array}\right], B=\left[\begin{array}{rrr} 3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 5 \end{array}\right] $$

Short answer

So, the result of the matrix multiplication AB is \( \left[\begin{array}{ccc} 3 & 0 & 0 \ 0 & -4 & 0 \ 0 & 0 & -10 \end{array}\right]\).

Step-by-step solution

Define Matrices A and B

Let's define Matrix \(A\) and Matrix \(B\) as: \(A = \left[\begin{array}{ccc} 1 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & -2 \end{array}\right]\) \(B = \left[\begin{array}{ccc} 3 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 5 \end{array}\right]\)

Perform Matrix Multiplication

Now, multiplying each element in the row of the first matrix with the corresponding element in the column of the second matrix, and summing them up to get the element at the respective position in the resultant matrix. \(AB = \left[\begin{array}{ccc} 1*3 & 0*(-1) & 0*5 \ 0*3 & 4*(-1) & 0*5 \ 0*3 & 0*(-1) & -2*5 \end{array}\right]\). Which results in: \(AB = \left[\begin{array}{ccc} 3 & 0 & 0 \ 0 & -4 & 0 \ 0 & 0 & -10 \end{array}\right]\).

Key concepts

Diagonal Matrix

A diagonal matrix is a special type of square matrix. In a diagonal matrix, all entries outside the main diagonal are zero. The main diagonal runs from the top left to the bottom right corner.
This design simplifies many matrix operations, including multiplication.

• Example: For a 3x3 diagonal matrix like \(A\), elements \( a_{12}, a_{13}, a_{21}, a_{23}, a_{31}, \) and \( a_{32} \) are all zero.
• Only \( a_{11}, a_{22}, \) and \( a_{33} \) (the diagonal elements) have non-zero values.

Diagonal matrices are crucial in various mathematical areas. They make it easy to understand the properties of matrices.
For instance, eigenvalues of diagonal matrices are simply the diagonal elements themselves.
Their simplicity allows for straightforward computation in matrix arithmetic as well.

Matrix Operations

Matrix operations involve addition, subtraction, and multiplication. Multiplying matrices is a bit more complex than the other operations.
When multiplying two matrices, you take the dot product of rows and columns.

• The number of columns in the first matrix must equal the number of rows in the second.
• The element in the \(i^{th}\) row and \(j^{th}\) column of the resulting matrix is computed by multiplying corresponding elements from the row of the first matrix and column of the second matrix, then summing up these products.

This method ensures proper alignment and multiplication for each matrix element. With diagonal matrices like \(A\) and \(B\) from our exercise, matrix multiplication becomes easier since zeros keep computations minimal outside the main diagonal.
For instance, multiplying two diagonal matrices results in another diagonal matrix.

Matrix Arithmetic

Matrix arithmetic includes operations like addition, subtraction, and multiplication. While addition and subtraction are straightforward, following element-by-element operations, multiplication requires additional rules:

• For two matrices to be multiplied, they must be compatible (i.e., the number of columns in the first matrix must equal the number of rows in the second).
• The resulting matrix's size is determined by the number of rows of the first and columns of the second matrix.

In our solved example, both matrices are 3x3, which makes them compatible for multiplication. With diagonal matrices like \(A\) and \(B\), computations are direct.
Non-diagonal elements multiply to zero, leaving only the diagonal elements to be multiplied directly.
This simplification is one of the key advantages of working with diagonal matrices in matrix arithmetic. It also speeds up computation significantly, which is beneficial for larger matrices.