![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCrcwMj7wX_bH3veKySPTlQMGiz6fWzfXlhviRwSzeT9bZWCq37VvQFkeG1_QXEc3JGrvHBaXjcp7zu8OuTB8COcsKG5ACh2qvhT6Fxcr6UBD4ox0Lrd58UUuQF3LaJd3eEJsts05KVJ30/s400/cube-matrix.jpg)
Picture is obtained from http://blog.iseesystems.com/wp-content/uploads/2009/07/cube-matrix.jpg
Solution:
(Explanation can be obtained from the Properties of Matrices, presented after the answer).
Rewriting the simultaneous equations,
2x + 5y = 39
11x - 3 y = 1
The simultaneous equations may be represented by the matrix equation
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRomGtUKI09FtmmrdVxYtZ8Ciha3Qy360plrl1aN-og2KOGKl0TZNBfgvzZYwaGyzvdmBZaVEiuNU5gb1jIAM-rtYKI__u5-AlMSbrPceBbn3HJwTLF7c4c53hoV6pfe00co7HDRx4VJSK/s400/Matrix+representation+-+1.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzjYZoJ0mS_JoZOGmV5CBY3d4b_sEnPvXxvoUuCKGOhwHbsIDQSjsN0QKqEk8iEUdg4O72MziNnBzouERYboA16wuGjILgmqMn1bAHrtWE06QKNzQC2lTck-HmZhmisBLPEUYgiAoW0F4L/s400/Determinant+of+Matrix+A+%28det+A%29+-+1.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-qXAdRaAynHR5twcZRKjjXFzJMuKoEKQFoMG1dJCAs8i2V2HO5TbsHYqcmmZ-fcbn8BciEd2254VyEq4aRsHMBf46nVXhVwNGOmlAQF2riyavlRpQMpmZO3YYqUpu5X7mVWPElnQT4rHT/s400/Inverse+Matrix+of+A.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOmOGEz_84vA1yUhOivnryC5Cl3Dyz06WQQSwZfeDy_AunQkykuY4oyXv5IdVhPgueNPZ-QP6pGS6QwZ-TIr4S0b_mBGexwgGidhRrgcKtfkzVPoDhLHjEIDRbCUPcI0Ly9XKwKg4Iz4f5/s400/Solving+for+X+-+1.jpg)
Therefore x = 2, y = 7 (Answer).
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhd-cPKUcocQ5ompCj1bSsnSNchzKKXyqQ-hY7cMh6LXM2UGq1hrv0buEurBKTit8XvU1av31NiQebA43pqEdtYd798Q4Fl7i-qjLo7lAI9LEjaP-jnH3ILFh_wMbMpjuAVKy8b_k6PQK_A/s400/Matrix1-1.jpg)
Picture is obtained from http://www.philipmantione.com/matrix1.jpg
Properties of Matrices
From: http://www.sosmath.com/matrix/matrix0/matrix0.html - Introduction and Basic Operations (Author: M.A. Khamsi)
Addition of Matrices:
- Matrices involved in the addition operation must have the same size
- To add two matrices, add the entries one by one
The following are obtained from Additional Mathematics, 8th Edition, by The Keng Seng, Loh Cheng Yee, Consultant: Dr. Yeap Ban Har, Shinglee Publishers Pte Ltd, Chapter 8 - “Matrix and its Applications”, Page 215 to 229.
Multiplication of Matrices:
- A matrix of the order m x n multiplied by a matrix of order n x p will result in a matrix of order m x p.
Given that
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLK1Z6OzRcN1mVsDIGj0P329kWikJLJtqEG8SOL9Nm5GHBtCfEriW1I-4fJPjge_-bOz_nhK7_b_vkXi6PFv7G6kTlSwm4Osn0CTPrgcnxnVmhBuQNuSwCuYbDGqDp_fz8Gw-2rgnsT07N/s400/Order+of+Matrix+A,+B+and+C.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEginzkJwZvA2OPKHwrhKci8QzovkGWWLMKOdVG4-y4gnrWpu9d2sXbl-0tjnQaC6EimOuaWSzRAxhjHzhAsBSma5veAEiQk60EzFZS5VrLwIG8JEstU7Lhzwf-TJ2Syd5DhuEzT8vI_aEmP/s400/Multiplication+of+Matrices+-+%28i%29.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9c0ffSOjnP903UgHhodTdTVHAd7oOufpEguMX9bdgPsFSMsu9H47azyMyZKHJ-VymgyLOd2jSWWsMziqz1T0DwcMGeP9dEGov2RiuMujySDgVQryEbFyGtWixMFdnr3x63Jc9Rc6tFfJw/s400/Multiplication+of+Matrices+-+%28ii%29.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjatWU3SJdKUbbTZ4B2GsWx9afVQx9tp0uKwyO9qlwTdBwG7TLNAhBi_RoGNlTSJjhUxsWsjI2D1fJ3ZuYmVSq93_xj45vc84ppM5A5uR1Rl6cSteFE0S5Z9sj00v8P5zTTCxSgRTSzFMXr/s400/Multiplication+of+Matrices+-+%28iii%29+and+%28iv%29.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLkYFf-PEI6aVTUydmCy0hROiyVu3GRew8z5xIE9D4RpyS5a5nG2Rzx2b02wm_ilzGhBFgSUTokGdEcnRrmJVSRei6iBZdoWv2q0ytCbyP-Xw1B8P9A3q6xDuC9BRSLvVPkCnmo_jYAtMR/s400/Multiplication+of+Matrices+-+%28v%29.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5Ud2TNmbk8uSdj05svxku51E62XFdO_Yhub6WmxK0nd2evy_-Qj6asKIdwdtdyGdfqyxHEEs5cKhimDXU4drM4LNbvTcKIrHZPtPfhEfMGGT6Oa28q0swB5eJKEO2qbNz2sPubQiCk0iV/s400/Multiplication+of+Matrices+-+%28vi%29.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFeu-x4bWMEWkjkkwht2YdcJvvFu8PrQa7QOEzsdydOnMWBFWCYQIKADFyXMbZlA4oQidn_LWoLUNQCTAsDIpEuC-3yqBaVQLCTDzqOept9ku6olsrtOenzW-17wZgCxUfr7TMA9ZTgtSS/s400/Note+for+AB+equal+BA.jpg)
The Inverse Matrix
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiUEuYTL6IfJw03Xq_hsLWZSc4vpVUGnWkTnOnWw68xRlroqk2pcPqDxSmkIwSbbM4NnoiDtov1VOf6tTSFaMNWbNsQ7UD7mJEMxwShO5f6yZYkYf4HCReuCjgqFzgQcGNUjGDA2OnmEmT/s400/Suppose+A.jpg)
considering only 2 x 2 matrices, we have:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpZtEXa3dcjC4Og7h3hkk9-4NkdNsiMa_FCmK3OUu0wiYgMebkkFdJgNLp1ihOjTT3sITqtD9U5KE9EuG6l2hbOQv5FE-HGDP_GhIZOgtMjuYTaQ9EDBuz3FgemvJcVNW1nkXezyW2wB0K/s400/Null+Matrices.jpg)
In the real number system, there exists an multiplicative identity, 1, such that a x 1 = 1 x a = a.
Similarly there exists for matrices, the multiplicative identity, I, such that AI = IA = A.
In the number system, for every number a, there is an additive inverse, -a, such that a + (-a) = 0;
and a multiplicative inverse,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheSeXEuIM4tkAqFsDQR2qiLVvUQ9dM3OBeOEmxpF4He2-wkyWtJ14XMYEIVRYvVzR7i18WvArxBQPkRK0RbWjYyGJnyqyJk2uJ7SUZHMt5_J90OcbrSavehnwag0CX8z6gU6BfS7UUKq13/s400/Multiplicative+Inverse+of+real+number+system.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2uCmOvBPwVp8vNQ02qnTDwcVOfLqzp23TgHhGBdEiwlPLUKL5YbfN0BPrkl0oEagUhvDAfXbCQyQK7tUB6_KwZFhAyOif3rZrFuBTxy2R8m-78WWNYvF57IgJHIiFRgDGP2Nt6ni_z-Dj/s400/Additive+Inverse.jpg)
is the additive inverse as
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgK-I86akuMh5DPKJ3EwZg4ioXhZk69gNncDn3UlJq3mxktUR18BMHOIY4ImnvIDBc1KNa1kwicY6vPXo9nHtwrQS9yshCcPU3zdXegTUUm6CXzv8ZTqM-TTN5oxwJlkfxg0GJp1Mwnn6CA/s400/Additive+Inverse+-+1.jpg)
The additive inverse of a matrix can be found easily.
The multiplicative inverses of matrices are the following.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicJi04Vl3LGrKAABnLbEZntnB_bTOnipfzEu24CPkujTj7h_lgmavt6fTcRNd8wpPm-i4fuZ775BB4fqldIfzvLu5MNPaLeSkMIbk0-Df_tNhdvk1a6K6DvA1v6cZiimCQgotca0roQhpw/s400/Multiplicative+Inverse+of+Matrices.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEg8y2V-itgsuoikWFpQx0NtgZqeSIn8wiBbyjbT1R9J_27qnKIPB_vSmx8EjzPkeW9Q0J-iKKlAbiz3hkCMiGnvVkOyMdW0mM4SmgxlWGfmvQMT0NsuhtrCr5LnVSUwlL9VSIjSndEVkp/s400/Inverse+of+A.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1R0hjX_T8fhL0zkM8xLtGHbd-EACQdqau9OPuXG0NESIRnAq8w0KEJ-OvE-bMZ2i5M3yLOlX4oHSEf8sPDXlOQfshQP2UELqCn6odmpNOrRvBkJbJoQ9LV8ArS9DOW_4Et2JBMLLLRaeI/s400/Inverse+of+B.jpg)
When the inverse of a matrix is mentioned, it generally refers to its multiplicative inverse.
If A is the inverse of B, then the order of A and B are the same, eg. 2 x 2.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTX4qx7Azvk_3XaovjZJfmfubnZ0oCkK5qBAlYwlqivscyXSqyrhqPawNrlRoXeBxlElEWirCycQ5jLwnfUbgiGQYrAJEyZ8JkOwmDShXiV3nv2sTTUDsqlqqtS1MPW_Bioo03tO5_08Rp/s400/Matrix+A.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUDvvZ8uP_nkN8OilIqA1uDSFDjbHghRvLvVpi7O-X859EEQ_57mqCKLmRmhsC1y8JKoYGNG6wUwlYJB48BZtD4GSqvZNJDs2ukdNDCiU0cvln6VDm4xg2em9dVdZECGiZw0CYN8yjeyWT/s400/Matrix+B.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhP-rvAr6g9LCKGw8bO8Tr4KsYrzhf1ygM-YHTVaGG0moybutb1jGb8yJhhBO_ZkDT_WKEU0-y1pQm34tenlO83rXZz0i64Z8lruoo48__6pvwRWqYhAJ1acShhM8MXRdEtCBibqZq2YPnh/s400/Product+of+AB.jpg)
Post-multiplying A by B, i.e. AB, we have,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz_SDO-bR_rAHc5m8bPjhCzoy2JJ9HfJnYihFYHPjUwDpQHgxd9EFwpKcm6NA7ENM0Scni8YwdpgCzLxZDZk3Ic6hyphenhyphen0sH6LTIcF8d-kANQLUkLgXsettt_sTgw8-O-pzzEEqcs2vcK2sS1/s400/Product+of+BA.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX5uDP4dKs9zxMhvL5gxgZ2m_LRWvipqpRO4ksDFXJyLQg85Htyi6vRc3GKGwhyHkF-eNDB-tUnqmPcM1jRgIWoScLKO0LfggD-2B0kTcTwY6fNjHLtH9EOE1XkZ0XR0AR4AhnRK-sEsqd/s400/Inverse+of+Matrix+A.jpg)
General Cases:
Consider the general 2 x 2 matrix
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4OvAdzxoNq1Gj6SLo1bK99KRumh7oh3DtRXb6qP__0ROjkpPeck-om02pQpXUSHN37565GPXh7z00AkWlHDK4xGiumANPQxFS239fmIEFo53SoXd_EzcAdu7UvJ1cL3xdkU5m9eKe-xa6/s400/Matrix+A+in+abcd+form.jpg)
Using the multiplicative inverse matrix rule above, let
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiY-wnxR_KANkIDzHjHxsJZUISNpkDRPhzpV_OEbHsUsBIMPYhmN0x2bf_fNrdX_65yf5OQsiMwlZqWYQZZ7wXIU-hxO1aYz_ofGMUEMRvjg8TFg0rJtiC1gOjgHmaaWpnWx2_ySaPqcnk/s400/Matrix+B+in+abcd+form.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyjOTyhJXPhxKBGnRiO91BbMcry6iSjCnM54o-H0YzKpMdBsOEbFa7OWf1UD6ME3elLImOu0FwGf43bgII10k6dhN_1Ziw-5sooPrRslnJHJfx7jq8YTOx_jHpArfakfRdoGkld-7Yav2Y/s400/Product+of+AB+and+BA+in+abcd+form.jpg)
In general,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5QBFC50QIokXEVUWgdTiNp6ev3r6R58fDrxj2gP25SFDpj2_J9fZwh0HLwVB4KSZbkNH1wqj5cMh7-JOGa_gdN_odv-J_HrrN8snMk_y2ZLAoxO5lsKZTL8znuy5Ju0XP9oUTFc8Fw6L7/s400/General+Case+of+inverse+of+a+matrix.jpg)
The expression ad - bc is known as the determinant of the matrix A and is denoted by
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPI-ZpuzKy1nLHtPfwbRj2w6KHJszWDnlDv1tGbhIHowvmJeyTZ1KafHfovCKR8eWnbz2ZmwQniIEx3kXKfZ7Uj5Kp0JQBzU_wMaJBJ5AQTF7Z4x_Zre38aLm0JaAEDQviAMU845JNqcQy/s400/Det+A.jpg)
If det A = 0, then the inverse of A is not defined because
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWYXZKrx823mR23Vb26gdVILIiVNpK1zeVeAS_pgHcSy0Yl4qLb43tahcH4JZVvCL0Pp6_Zb__0j5zDUIypaJNyRI8qz1FGpbSgHN9Dw8bkbT7v5ee795bsu-xg3gWPUCOQbzzx3Ua-n7k/s400/1+divided+by+0+%28not+defined%29.jpg)
In such a case, A does not have an inverse.
Where a matrix does not possess an inverse, it is known as a singular matrix.
References
- Additional Mathematics, 8th Edition, by The Keng Seng, Loh Cheng Yee, Consultant: Dr. Yeap Ban Har, Shinglee Publishers Pte Ltd, Chapter 8 - “Matrix and its Applications”, Page 215 to 229.
- http://www.sosmath.com/matrix/matrix0/matrix0.html - Introduction and Basic Operations (Author: M.A. Khamsi)
- http://www.sosmath.com/matrix/matrix1/matrix1.html - Multiplication of Matrices (Author: M.A. Khamsi)
- http://blog.iseesystems.com/wp-content/uploads/2009/07/cube-matrix.jpg
- http://blog.iseesystems.com/tag/ithinkstella/
- http://www.philipmantione.com/matrix1.jpg