Question

in the table shown, ( i ) represents the imaginary unit. select all cells in the table for which the product of the row value and the column value is a real number. select all appropriate cells in the table.

Explanation:

Step1: Recall the property of \(i\)

We know that \(i^2=- 1\).

Step2: Calculate the products

• For the row \(i\):
• \(i\times(-3)=-3i\) (not a real number).
• \(i\times(-2i)=-2i^2\). Since \(i^2 = - 1\), then \(-2i^2=-2\times(-1) = 2\) (a real number).
• \(i\times5 = 5i\) (not a real number).
• \(i\times i^2=i^3=i^2\times i=-i\) (not a real number).
• For the row \(-4\):
• \(-4\times(-3)=12\) (a real number).
• \(-4\times(-2i)=8i\) (not a real number).
• \(-4\times5=-20\) (a real number).
• \(-4\times i^2=-4\times(-1) = 4\) (a real number).

Answer:

For the row \(i\) and column \(-2i\); for the row \(-4\) and column \(-3\); for the row \(-4\) and column \(5\); for the row \(-4\) and column \(i^2\)