The first two columns of M−1 contain the essence of the definition of the second, s (adopted in 1967) and the meter, m (adopted in 1983), as shown in the box, where it is concluded that the definitions of the second and the meter did not change in 2019. The kg column defines the kilogram to be 1 kg . The proportionality relation becomes an hv / c 2 Cs equality by dividing each quantity on the right-hand side by its exact numerical value [5], which is given in Table 1. The numerical factor (truncated here for convenience) is: 1 kg 1.476 10 hv / c2 Cs 40 (7) This is the definition of the kilogram found in section 2.3.1 of [2]. The definition is realized by methods where the mass of an object is determined by traceability to the three defining constants shown in (7) [2], [10]. Note that the matrix technique defines all base units simultaneously [5]. Matrix M−1 Take the top-left corner of Fig. 1 as a simple example of using matrices to define the SI second and meter. s^ 1 1 M m^ 0 1 vcCs s First calculate the inverse of M using a function such as MINVERSE in Excel: m M1 ^1 1 c ^ 01 v Cs can also be used to find the definition of any SI unit that can be written in terms of the subset of four chosen base units (1) [8]. The same results are obtained if, for example, the joule (J) had been chosen as a base unit instead of the kilogram. The definitions of the kilogram and all other units are exactly the same even when the joule functions as a " base " unit from which the kilogram definition is " derived. " However, the inverse of the matrix M is still required to define all SI units. Only square matrices can be inverted, but not all square matrices can be inverted. Since it is the inverted matrix that is used to define all SI units in terms of the selected base units, this places an automatic constraint on any selection of base units. Aside from that, any set of base units can be chosen. When the joule plays the role of a base unit in Fig. 1, the column of labels substitutes J^ for kg^. Therefore, the third column of the yellow matrix contains the exponents in the product s1 m0J1 A0 because the Planck constant has the unit J s, as shown in Table 1. After the matrix is inverted, the joule remains defined by J hv Cs and (7) remains the definition of the kilogram. Although the choice of base units is not unique, the definitions of all SI units defined by Table 1 are indeed unique. To use matrix M−1 of Fig. 2 to define any arbitrary unit that can be written in terms of the chosen base units (normally the historical base units) [8], one begins with the desired product of powers of base units as shown in (1). The exponents are written in a column. An example will make the process clear. In Table 4 of [2], the ohm (symbol: Ω) appears as a special name for the following product of four base units: Ω kg ms A 23 2 (8) which is another way of expressing that ampere squared, multiplied by the ohm, equals the watt. February 2023 The yellow and gray matrices are identical in this example, but the meaning of the columns and rows is not the same. In general, an inverse matrix is different from the starting matrix. The difference between Fig. 4 and Fig. 5 provides a striking example. By matrix multiplication [9], which is implemented in Excel by the MMULT function, it can be verified that the product of M and its inverse M−1 MM 1 11 11 1 01 01 0 is the identity matrix, I: 1 I The inverse matrix is used for the next steps; next steps would be impossible if M could not be inverted. Column 1 of M−1 1 s , from which the SI second is defined to be = 9 192 631 770/ΔvCs Cs 1 s / Cs SI }SI Cs . This is the definition of the second found in [2] and is numerically identical to the 1967 definition (one second equals {ΔvCs ticks of a cesium atomic clock). Column 2 of M−1 1 , from which the meter is defined to be 1 m = c/{c}SI·{ΔvCs m Cs c 1 }SI /ΔvCs ≈ (30.663 319)·c/ΔvCs . This is the definition of the meter found in [2] and is numerically identical to the 1983 definition (one meter equals the distance light travels in a vacuum during 1/{c}SI seconds). The column matrix, V, of the various exponents is shown in gray in Fig. 3, where the labels to the left of V are a reminder of the source of the exponents appearing in (8). Simple matrix multiplication is then required to obtain the definition of one ohm. This can either be done by hand [9] or using packaged software-for example, the function MMULT in Excel. The matrix multiplication M−1 ·V results in powers of the four defining constants in the order they appeared in the columns of M−1 . The result is: Cs again h and e2 {h}SI and e , respectively, to establish equality to the ohm. In this case, the natural quantity h/e2 2 SI von Klitzing constant, symbol: RK 2 IEEE Instrumentation & Measurement Magazine has a name and symbol: the [6], [7]. Its SI value is now exact but is approximately he/ 25813 Ω. The definition of 9 v c he he 00 1 2 / Ω, where 2 must be divided by their exact numerical values contains the information 1 contains the information

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