How to Solve Complex Numbers, Matrices & Vectors on a Scientific Calculator

Step 1.1: Activating Complex Mode

To enter the complex calculation environment, press the MODE key, inspect the numeric options on your LCD display, and press 2 (labeled CMPLX). The top status bar will immediately display the small CMPLX indicator. In this mode, the imaginary unit $i$ is activated on the ENG key.

Step 1.2: Entering Rectangular Numbers ($a + bi$)

To enter a complex quantity like $12 + 5i$, type:

The calculator displays 12+5i. To toggle between the real component and the imaginary component on single-line displays, press SHIFT followed by =.

Step 1.3: Direct Rectangular to Polar Phasor Conversion ($r\angle\theta$)

Converting Cartesian coordinates to polar phasors is mandatory for AC power factor and voltage phase lag calculations. With your calculated answer on the screen:

1. Press SHIFT followed by 2 (the CMPLX menu).
2. Select option 3 (labeled ►r∠θ).
3. Press =.

For $12 + 5i$, the calculator instantly returns $r = 13$ and $\theta = 22.6198^\circ$. If your calculator is in Radian mode (SHIFT SETUP 4), $\theta$ will automatically be evaluated in radians.

Step 1.4: Direct Polar Phasor Entry ($r\angle\theta$)

You can directly input polar phasors using the angle symbol ∠, located above the negative sign key (-). To enter a load voltage of $220\angle-30^\circ$:

To immediately convert this polar phasor back into rectangular real and imaginary components ($a + bi$):

Step 1.5: AC Parallel Impedance Calculation Example

Calculate the equivalent impedance of two parallel branches: $Z_1 = 50 + 20i\ \Omega$ and $Z_2 = 30 - 40i\ \Omega$. Using the standard formula $Z_{eq} = \frac{Z_1 \cdot Z_2}{Z_1 + Z_2}$:

Result: $Z_{eq} = 32.35 - 3.24i\ \Omega$. Converting to polar via SHIFT 2 3 yields $32.51\angle -5.71^\circ\ \Omega$. This saves 15 minutes of manual pencil-and-paper rationalization!

Step 2.1: Defining and Populating a Matrix

To initialize and store a $3 \times 3$ matrix into $[MatA]$:

1. Press MODE followed by 6 (MATRIX).
2. Select 1 to select MatA.
3. Select 1 for dimension $3 \times 3$ (or 2 for $3 \times 2$, etc.).
4. Enter each value into the matrix grid, pressing = after each element to advance to the next cell.
5. When all 9 elements are entered, press the AC key. Do not panic! The data remains safely stored in the non-volatile $[MatA]$ register.

Step 2.2: Calculating Matrix Determinant ($\det(A)$)

The determinant reveals whether a matrix is invertible ($\det(A) \neq 0$) and provides the scaling factor for geometric coordinate transformations:

1. Press SHIFT followed by 4 (the MATRIX secondary menu).
2. Press 7 to select det(.
3. Press SHIFT followed by 4, then select 3 (MatA).
4. Close parentheses ) and press =.

Step 2.3: Inverting a 3x3 Matrix ($[MatA]^{-1}$)

Finding the inverse of a $3 \times 3$ matrix by hand requires calculating 9 minors, applying a cofactor sign matrix, transposing the adjugate, and dividing by the determinant. On your scientific calculator:

1. Recall $[MatA]$ to the screen: SHIFT 4 3.
2. Press the inverse key x⁻¹ (do not use the $\wedge -1$ exponent key).
3. Press =.

The screen displays the inverted matrix $[MatAns]$. You can scroll with the arrow replay pad to inspect fractional or decimal values for all 9 cells.

Step 2.4: Solving 3 Simultaneous Linear Equations

Given the system of equations:

Express in matrix notation $[A][X] = [B]$, where $[A]$ is the $3 \times 3$ coefficient matrix and $[B]$ is the $3 \times 1$ column matrix of constants:

1. Store coefficients into $[MatA]$ ($3 \times 3$).
2. Store constants $[8, -11, -3]^T$ into $[MatB]$ ($3 \times 1$) via SHIFT 4 1:Dim 2:MatB 3:3x1.
3. Clear screen with AC.
4. Execute the matrix inversion product:
   MatA [x⁻¹] × MatB =

The display immediately yields $[X] = [2, 3, -1]^T$, confirming $x = 2$, $y = 3$, $z = -1$.

Step 3.1: Defining 3D Vectors

To enter 3D Cartesian coordinates for $\vec{A} = 3\hat{i} + 4\hat{j} - 2\hat{k}$:

1. Press MODE 8 (VECTOR).
2. Press 1 (VctA) and select dimension 1:3.
3. Input coordinates: 3 = 4 = -2 =.
4. Press AC to store.

Repeat for $\vec{B} = 1\hat{i} - 2\hat{j} + 5\hat{k}$ by pressing SHIFT 5 (VECTOR) $\rightarrow$ 1:Dim $\rightarrow$ 2:VctB $\rightarrow$ 1:3.

Step 3.2: Dot Product ($\vec{A} \cdot \vec{B}$) & Work Done

The scalar dot product yields the projection of one vector along another. In physics, Work $W = \vec{F} \cdot \vec{d}$:

1. Recall VctA: SHIFT 5 3.
2. Insert the Dot operator: SHIFT 5 7:Dot.
3. Recall VctB: SHIFT 5 4.
4. Press =.

Step 3.3: Cross Product ($\vec{A} \times \vec{B}$) & Torque

The cross product generates a vector orthogonal to both operands. In statics, Torque $\vec{\tau} = \vec{r} \times \vec{F}$:

The display outputs $[VctAns] = [16, -17, -10]^T$, representing $16\hat{i} - 17\hat{j} - 10\hat{k}$.

Step 3.4: Finding Vector Magnitude ($|A|$)

To determine the Euclidean norm (magnitude) of $\vec{A}$:

Bitwise Operations (AND, OR, XOR, NOT)

In Base-N mode, pressing SHIFT followed by 3 (LOGIC) opens bitwise logic functions:

• and: Masks bit fields (e.g., 0b1101 and 0b1011 = 0b1001).
• or: Sets specific register flags.
• xor: Toggles bits or evaluates parity difference.
• Not: Two's complement bitwise inversion.