Problem 38 - Calculus for Biology and Medicine

Question

Let \(A=\left[\begin{array}{rrr}1 & 3 & 0 \\ 0 & 0 & -2 \\ -1 & 1 & 1\end{array}\right]\) and \(I_{3}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\) Show that \(A I_{3}=I_{3} A=A\).

Step-by-Step Solution

Answer

Multiplying a matrix by the identity matrix leaves it unchanged, so \( A I_3 = I_3 A = A \).

Step 1: Determine Dimensions

First, recognize the dimensionality of both matrices. Matrix \( A \) is a \( 3 \times 3 \) matrix and \( I_3 \) is the \( 3 \times 3 \) identity matrix. This means that both \( A I_3 \) and \( I_3 A \) are valid matrix multiplications, resulting in \( 3 \times 3 \) matrices.

Step 2: Multiply \( A \) by \( I_3 \)

Compute the product \( A I_3 \):Each element in the resulting matrix relates to a dot product between rows of \( A \) and columns of \( I_3 \). Since \( I_3 \) is the identity matrix, multiplying any matrix by \( I_3 \) on the right leaves the original matrix unchanged.Thus, \( A I_3 = A \).

Step 3: Multiply \( I_3 \) by \( A \)

Compute the product \( I_3 A \):Each element in the resulting matrix relates to a dot product between rows of \( I_3 \) and columns of \( A \). Since \( I_3 \) is the identity matrix, multiplying any matrix by \( I_3 \) on the left leaves the original matrix unchanged.Thus, \( I_3 A = A \).

Step 4: Conclusion

From Steps 2 and 3, we see that multiplying \( A \) by the identity matrix \( I_3 \), whether on the right or the left, results in the matrix \( A \) itself. Therefore, it is shown that \( A I_3 = I_3 A = A \).

Key Concepts

Identity Matrix

An identity matrix is a special kind of square matrix. It is noted to have ones on the main diagonal from the top left to the bottom right, and zeros everywhere else. The identity matrix is like the number 1 in regular arithmetic - multiplying anything by 1 doesn’t change its value. In matrix multiplication, this principle applies similarly. This is why it's notable:

• Multiplying a matrix by an identity matrix, either on the left or the right, leaves the original matrix unchanged.
• For a square matrix, say, of size 3x3, like our matrix in the exercise, the identity matrix must also be 3x3 (such as the matrix denoted as \(I_{3}\)).
• In our example, for matrix \(A\), multiplying \(A\) by \(I_{3}\) on either side just results in \(A\) again: \(AI_{3} = A\) and \(I_{3}A = A\).

Whether it's left-multiplying by the identity matrix or right-multiplying, the result is your original matrix. This characteristic is fundamental when considering matrix operations.

Dot Product

The dot product serves as a building block for understanding matrix multiplication. In the context of matrices, it's used to compute elements when multiplying matrices together. For each element of the resulting matrix:

• It represents the sum of the products of the corresponding entries from a row of the first matrix and a column of the second matrix.
• Specifically, in our setup, we calculate the product of rows from matrix \(A\) and columns from the identity matrix \(I_{3}\).
• For instance, if calculating the element in the first row and first column of the product of \(A\) and \(I_{3}\), take the dot product of the first row of \(A\) with the first column of \(I_{3}\).

Because the identity matrix has zeros everywhere except the diagonal, many products in the dot product calculations will involve multiplying by zero, thus not affecting the original values. This is why multiplying by an identity matrix results in the original matrix.

Matrix Dimensions

Understanding the dimensions of a matrix is crucial when handling matrix multiplication. The dimensions are given in terms of rows and columns, written as \(m\times n\), where \(m\) is the number of rows and \(n\) is the number of columns.

• Each dimension must agree appropriately for matrix multiplication to proceed: the number of columns in the first matrix must equal the number of rows in the second matrix.
• In our example, matrix \(A\) is a \(3 \times 3\) matrix, and \(I_3\) is also a \(3 \times 3\) matrix.
• When they are multiplied, the resulting matrix is of size \(3 \times 3\), demonstrating that the operation is well-defined and can proceed.

Always check dimensions first to ensure that your matrices can indeed be multiplied, as a mismatch in dimensions will make multiplication impossible. This step is foundational in matrix arithmetic.