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RPN Power 48 manual · Chapter 12 of 19

Vectors and matrices

Type vectors and matrices, take determinants and inverses, multiply, solve linear systems, and use dot and cross products.

12.1Type a vector or matrix#

Left shift 9 gives square brackets. A vector is one pair of brackets with numbers separated by spaces. A matrix is a pair of brackets around one inner pair per row. The easiest way to type a matrix: press left shift 9 twice, type the first row, press right shift ▶ to step over the inner bracket, press left shift 9 again for the next row, and ENTER.

Example 12.1

Type a vector

  1. Left shift 9, then 1 2 3, ENTER.

    ◀[ ]91SPC2SPC3ENTER
Result of the example "Type a vector" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: [ 1 2 3 ].
You should see[ 1 2 3 ]
Example 12.2

Type a 2 by 2 matrix

  1. Two bracket pairs for the rows.

    ◀[ ]9◀[ ]91SPC2▶▶◀[ ]93SPC4ENTER
Result of the example "Type a 2 by 2 matrix" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: [[ 1 2 ] [ 3 4 ]].
You should see[[ 1 2 ] [ 3 4 ]]

12.2The MATRX menu#

MATRX page 1: →ARRY, ARRY→, DET, INV, TRN, RREF. The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: / →ARRY ARRY→ DET INV TRN RREF .
MATRX page 1: →ARRY, ARRY→, DET, INV, TRN, RREF.
MATRX page 2: CON, IDN, SIZE, DOT, CROSS, ABS. The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: / CON IDN SIZE DOT CROSS ABS .
MATRX page 2: CON, IDN, SIZE, DOT, CROSS, ABS.
MATRX page 3: →V2, →V3, V→, GET, PUT, NEG. The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: / →V2 →V3 V→ GET PUT NEG .
MATRX page 3: →V2, →V3, V→, GET, PUT, NEG.
CommandWhat it does
→ARRY, ARRY→Build an array from stack objects, or explode it.
DETDeterminant of a square matrix.
INVInverse.
TRNTranspose.
RREFReduce to row-echelon form.
CON, IDNConstant matrix, identity matrix.
SIZEThe dimensions as a list.
DOT, CROSSDot and cross product of vectors.
ABSLength of a vector.
→V2, →V3, V→Make a 2D or 3D vector from numbers, or explode one.
GET, PUTRead or replace one element.

12.3Determinant, inverse and transpose#

det⁡(1234)=−2,A−1=(−211.5−0.5)\det\begin{pmatrix}1&2\\3&4\end{pmatrix}=-2,\quad A^{-1}=\begin{pmatrix}-2&1\\1.5&-0.5\end{pmatrix}
Example 12.3

Determinant

  1. Press DET (third key).

    ◀[ ]9◀[ ]91SPC2▶▶◀[ ]93SPC4ENTER◀MATRXVARF3
Result of the example "Determinant" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: -2 / →ARRY ARRY→ DET INV TRN RREF .
You should see1: -2
Example 12.4

Inverse

  1. Press INV (fourth key).

    ◀[ ]9◀[ ]91SPC2▶▶◀[ ]93SPC4ENTER◀MATRXVARF4
Result of the example "Inverse" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: [[ -2 1 ] [ 1.5 -.5 ]] / →ARRY ARRY→ DET INV TRN RREF .
You should see[[ -2 1 ] [ 1.5 -.5 ]]
Example 12.5

Transpose

  1. Press TRN (fifth key).

    ◀[ ]9◀[ ]91SPC2▶▶◀[ ]93SPC4ENTER◀MATRXVARF5
Result of the example "Transpose" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: [[ 1 3 ] [ 2 4 ]] / →ARRY ARRY→ DET INV TRN RREF .
You should see[[ 1 3 ] [ 2 4 ]]
Example 12.6

Size

  1. SIZE is the third key of page 2.

    ◀[ ]9◀[ ]91SPC2▶▶◀[ ]93SPC4ENTER◀MATRXVARNXTF3
Result of the example "Size" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: { 2 2 } / CON IDN SIZE DOT CROSS ABS .
You should see{ 2 2 }
Example 12.7

Identity matrix

  1. Type 2 and press IDN (page 2, second key).

    2◀MATRXVARNXTF2
Result of the example "Identity matrix" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: [[ 1 0 ] [ 0 1 ]] / CON IDN SIZE DOT CROSS ABS .
You should see[[ 1 0 ] [ 0 1 ]]

See also Enter and multiply matricesMatrix determinant and inverse

12.4Multiply matrices and solve systems#

Use +, − and × on matrices of matching size. Adding or subtracting a number and a matrix is Bad Argument Type; multiply or divide by a number to scale. A matrix times a vector gives a vector. ÷ with a vector on level 2 and a matrix on level 1 solves the linear system.

Ax=b  ⇒  x=b÷AA\mathbf{x}=\mathbf{b}\;\Rightarrow\;\mathbf{x}=\mathbf{b}\div A
Example 12.8

Square a matrix

  1. Copy the matrix with ENTER and multiply.

    ◀[ ]9◀[ ]91SPC2▶▶◀[ ]93SPC4ENTERENTER×
Result of the example "Square a matrix" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: [[ 7 10 ] [ 15 22 ]].
You should see[[ 7 10 ] [ 15 22 ]]
Example 12.9

Scale a matrix

  1. Multiply by 2.

    ◀[ ]9◀[ ]91SPC2▶▶◀[ ]93SPC4ENTER2×
Result of the example "Scale a matrix" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: [[ 2 4 ] [ 6 8 ]].
You should see[[ 2 4 ] [ 6 8 ]]
Example 12.10

Multiply a matrix and a vector

  1. Type the vector [ 1 0 ] and multiply.

    ◀[ ]9◀[ ]91SPC2▶▶◀[ ]93SPC4ENTER◀[ ]91SPC0ENTER×
Result of the example "Multiply a matrix and a vector" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: [ 1 3 ].
You should see[ 1 3 ]
Example 12.11

Solve a linear system

x+2y=53x+4y=6\begin{aligned}x+2y&=5\\3x+4y&=6\end{aligned}

  1. Type the right-hand side [ 5 6 ] first, then the coefficient matrix, then divide.

    ◀[ ]95SPC6ENTER◀[ ]9◀[ ]91SPC2▶▶◀[ ]93SPC4ENTER÷
Result of the example "Solve a linear system" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: [ -4 4.5 ].
You should see[ -4 4.5 ]

12.5Dot product, cross product and length#

a⋅b=a1b1+a2b2+a3b3\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2+a_3b_3
Example 12.12

Dot product

  1. Type [ 1 2 3 ] and [ 4 5 6 ], press DOT (page 2, fourth key).

    ◀[ ]91SPC2SPC3ENTER◀[ ]94SPC5SPC6ENTER◀MATRXVARNXTF4
Result of the example "Dot product" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: 32 / CON IDN SIZE DOT CROSS ABS .
You should see1: 32
Example 12.13

Cross product

  1. CROSS is the fifth key.

    ◀[ ]91SPC2SPC3ENTER◀[ ]94SPC5SPC6ENTER◀MATRXVARNXTF5
Result of the example "Cross product" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: [ -3 6 -3 ] / CON IDN SIZE DOT CROSS ABS .
You should see[ -3 6 -3 ]
Example 12.14

Length of a vector

  1. Type [ 3 4 ], press ABS (page 2, sixth key).

    ◀[ ]93SPC4ENTER◀MATRXVARNXTF6
Result of the example "Length of a vector" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: 5 / CON IDN SIZE DOT CROSS ABS .
You should see1: 5

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