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Matrix addition and scalar multiplication

\[ A = \begin{bmatrix} -3 & -3 & 3 \\ 1 & 0 & 2 \\ 0 & -2 & 3 \end{bmatrix} \]

\[ B = \begin{bmatrix} -1 & 3 & 3 \\ -2 & 5 & 2 \\ 1 & 2 & 4 \end{bmatrix} \]

\[ C = \begin{bmatrix} -5 & 3 & 9 \\ -3 & 10 & 6 \\ 2 & 2 & 11 \end{bmatrix} \]

In R:

A <- matrix(c(-3, -3, 3,
              1, 0, 2,
              0, -2, 3),
            nrow = 3, byrow = TRUE)
B <- matrix(c(-1, 3, 3,
              -2, 5, 2,
              1, 2, 4),
            nrow = 3, byrow = TRUE)
C <- matrix(c(-5, 3, 9,
              -3, 10, 6,
              2, 2, 11),
            nrow = 3, byrow = TRUE)

Find \((A - B) + C\):

(A - B) + C
     [,1] [,2] [,3]
[1,]   -7   -3    9
[2,]    0    5    6
[3,]    1   -2   10

Find \(2A + B\):

2 * A + B
     [,1] [,2] [,3]
[1,]   -7   -3    9
[2,]    0    5    6
[3,]    1   -2   10

They are equal, therefore:

\[ 2A + B = (A - B) + C \]

\[ A + 2B -C = 0\]

A + 2 * B - C
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0

sessionInfo()
R version 4.3.3 (2024-02-29 ucrt)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 11 x64 (build 22631)

Matrix products: default


locale:
[1] LC_COLLATE=English_United States.utf8 
[2] LC_CTYPE=English_United States.utf8   
[3] LC_MONETARY=English_United States.utf8
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.utf8    

time zone: America/New_York
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] workflowr_1.7.1

loaded via a namespace (and not attached):
 [1] vctrs_0.6.5       httr_1.4.7        cli_3.6.2         knitr_1.45       
 [5] rlang_1.1.3       xfun_0.42         stringi_1.8.3     processx_3.8.3   
 [9] promises_1.2.1    jsonlite_1.8.8    glue_1.7.0        rprojroot_2.0.4  
[13] git2r_0.33.0      htmltools_0.5.7   httpuv_1.6.14     ps_1.7.6         
[17] sass_0.4.8        fansi_1.0.6       rmarkdown_2.26    jquerylib_0.1.4  
[21] tibble_3.2.1      evaluate_0.23     fastmap_1.1.1     yaml_2.3.8       
[25] lifecycle_1.0.4   whisker_0.4.1     stringr_1.5.1     compiler_4.3.3   
[29] fs_1.6.3          pkgconfig_2.0.3   Rcpp_1.0.12       rstudioapi_0.15.0
[33] later_1.3.2       digest_0.6.34     R6_2.5.1          utf8_1.2.4       
[37] pillar_1.9.0      callr_3.7.5       magrittr_2.0.3    bslib_0.6.1      
[41] tools_4.3.3       cachem_1.0.8      getPass_0.2-4