HP 15c Manual

 Section 12: Calculating with Matrices 151 
 
Example: Calculate the transpose of matrix B. Matrix B was set in 
preceding examples to 
 
 
Keystrokes Display 
l > B b 2 3 Displays descriptor of 
2×3 matrix B. 
´ > 4  b 3 2 Descriptor of 3×2 
transpose. 
Matrix B (which you can view using l B in User mode) is now 
 
Scalar Operations 
Scalar  operations  perform  arithmetic  operations  between  a  scalar  (that  is,  a 
number)  and  each  element  of  a  matrix.  The  scalar  and  the  descriptor  of...

152 Section 12: Calculating with Matrices 
 
 
Operation 
Elements of Result Matrix* 
Matrix in Y-Register Scalar in Y-Register 
Scalar in X-Register Matrix in X-Register 
+ Adds scalar value to each matrix element. 
* Multiplies each matrix element by scalar value. 
- Subtracts scalar value 
from each matrix 
element. 
Subtracts each matrix 
element from scalar value. 
÷ Divides each matrix 
element by scalar value. 
Calculates inverse of matrix 
and multiplies each element 
by scalar value. 
* Result...

 Section 12: Calculating with Matrices 153 
 
 
Keystrokes Display 
1 - b 2 3 Subtracts 1 from the elements of 
matrix B and stores those values in the 
same elements of B. 
 
The result (which you can view using lB in User mode) is 
. 
Arithmetic Operations 
With matrix descriptors in both the X- and Y-registers, pressing + or 
- calculates the sum or difference of the matrices. 
Pressing Calculates* 
+ Y + X 
- Y - X 
* Result is stored in result matrix. 
Result matrix may be X or Y 
Example: Calculate...

154 Section 12: Calculating with Matrices 
 
 
Keystrokes 
 
Display 
   
- C 2 3 Calculates B - A and stores 
values in redimensioned result 
matrix C. 
The result is  
Matrix Multiplication 
With  matrix  description  in  both  the  X- and  Y-registers,  you  can  calculate 
three  different  matrix  products.  The  table  below  shows  the  results  of  the 
three  functions  for  a  matrix X specified  in  the X-register  and  a  matrix Y 
specified  in  the  Y-register.  The  matrix X-1 is  the...

 Section 12: Calculating with Matrices 155 
 
For ÷,  the  matrix  specified  in  the  X-register  is  replaced  by  its LU 
decomposition.  The ÷ function  calculates X–1Y using  a  more  direct 
method than does ∕ and *, giving the  result faster and  with improved 
accuracy. 
Example: Using  matrices A and B from  the  previous  example,  calculate 
C = AT B. 
 
Keystrokes Display 
l>
A 
A 2 3 Recalls descriptor for matrix A. 
l>
B 
b 2 3 Recalls descriptor for matrix B 
into X-register, moving matrix...

156 Section 12: Calculating with Matrices 
 
Solving the Equation AX = B 
The ÷ function is useful for solving 
matrix  equations  of  the  form AX  =  B, 
where A is the  coefficient  matrix, B is 
the  constant  matrix,  and X is  the 
solution  matrix.  The  descriptor  of  the 
constant matrix B should be entered in 
the Y-register and the descriptor of the 
coefficient matrix A should be entered 
in  the  X-register  Pressing ÷ then 
calculates the solution X=A-1B.* 
Remember  that  the ÷ function...

 Section 12: Calculating with Matrices 157 
 
 
 Week 
1 2 3 
Total Weight (kg) 274 233 331 
Total Value $120.32 $112.96 $151.36 
Silas  knows  that  he  received  $0.24 per  kilogram  for  his  cabbage  and  $0.86 
per  kilogram  for  his  broccoli.  Use  matrix  operations  to  determine  the 
weights of cabbage and broccoli he delivered each week. 
Solution: Each  weeks  delivery  represents  two  linear  equations  (one  for 
weight  and  one  for  value)  with  two  unknown  variables  (the  weights...

158 Section 12: Calculating with Matrices 
 
 
Keystrokes Display  
274 OB 274.0000 Stores b11.* 
233 OB 233.0000 Stores b12. 
331 OB 331.0000 Stores b13. 
120.32 OB 120.3200 Stores b21. 
112.96 OB 112.9600 Stores b22. 
151.36 OB 151.3600 Stores b23. 
´< Á 151.3600 Designates matrix D as result 
matrix. 
l> B b 2 3 Recalls descriptor of constant 
matrix. 
l> A A 2 2 Recalls descriptor of coefficient 
matrix A into X-register, moving 
descriptor of constant matrix B 
into Y-register. 
÷ d 2 3 Calculates...

 Section 12: Calculating with Matrices 159 
 
Silas deliveries were: 
 Week 
 1 2 3 
Cabbage (kg) 186 141 215 
Broccoli (kg) 88 92 116 
Calculating the Residual 
The HP-15C enables you to calculate the residual, that is, the matrix 
Residual = R–YX 
where R is  the  result  matrix  and X and Y are  the  matrices  specified  in  the 
X- and Y-registers. 
This  capability  is  useful,  for  example,  in  doing  iterative  refinement  on  the 
solution  of  a  system  of  equations  and  for  linear...

160 Section 12: Calculating with Matrices 
 
Using Matrices in LU Form 
As noted  earlier,  two  matrix  operations  (calculating  a  determinant  and 
solving  the  matrix  equation  (AX  =  B)  create  an LU decomposition  of  the 
matrix  specified  in  the  X-register.  The  descriptor  of  such  a  matrix  has  two 
dashes  following  the matrix  name. A matrix  in LU form  has  elements  that 
differ from the elements of the original matrix. 
However, the descriptor for a matrix in LU form can be...