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I suggest asking separate questions for complex numbers, and for matrices. Complex numbers are used in a variety of fields, one of them is electrical engineering. As soon as AC circuits are analyzed, it turns out that complex numbers are the natural way to do this.
Engineering (especially electrical engineering), Math, Physics mainly.
A complex number is any number that can be represented in the form of a+bi, the real numbers are a and b, the imaginary number is i. Complex numbers are used in scientific and engineering fields.
They are frequently used in Engineering applications.
The imaginary axis is used in the definition of the complex numbers. Complex numbers are used in many fields in engineering, in particular - electric engineering, aerodynamics, acoustics etc.
Mainly engineering, particularly control theory and other electronic systems; complex numbers are useful to represent oscillation frequency and phase information.
Various engineering fields will use them. Electricity and electronics.
I suggest you read the Wikipedia article con complex numbers, specifically the section "Applications". One example is electrical engineering: in the case of AC, it helps to express all voltages, currents and impedances (equivalent of resistances) as complex numbers.
Physics (e.g., quantum mechanics, relativity, other subfields) makes use of imaginary numbers. "Complex analysis" (i.e., calculus that includes imaginary numbers) can also be used to evaluate difficult integrals and to perform other mathematical tricks. Engineering, especially Electrical Engineering makes use of complex and imaginary numbers to simplify analysis of some circuits and waveforms.
As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics.
They are used for working out equations where the numbers you are working with are not physically possible, but we just imagine they are, such as the square root of a negative number In engineering, especially Electrical Engineering, using complex numbers to represent signals (rather than sines and/or cosines) make comparing and working with signals easier.
Complex math covers how to do operations on complex numbers. Complex numbers include real numbers, imaginary numbers, and the combination of real+imaginary numbers.